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Which measure of dispersion is the best and how?
Standard deviation is the best measures of dispersion, because it posseses most of the characterstics of an ideal measure of dispersion.
Some measures of dispersion depend upon the spread of values whereas some calculate the variation of values from a central value. Do you agree?
Yes we agree with the statement Range and quartile deviation measure the dispersion by calculating the spread within which the values lie i.e. they depend on the spread of values. On the other hand, mean deviation and standard deviation calculate the variation of value from a central value.
In a town 25% of the persons earned more than Rs. 45000, whereas 75% earned more than Rs. 18000. Calculate the absolute and relative values of dispersion.
Absolute value of dispersion i.e. 
A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are:
|
X |
Y |
|
25 |
50 |
|
85 |
70 |
|
40 |
65 |
|
80 |
45 |
|
120 |
80 |
Which batsman should be selected if we want
(i) a higher run getter, or
(ii) a more reliable batsman in the team?
Batsman -A
|
Scores (X) |
|
|
|
25 |
–45 |
2025 |
|
85 |
+15 |
225 |
|
40 |
–30 |
900 |
|
80 |
+10 |
100 |
|
120 |
+50 |
2500 |
|
ΣX= 350 |
Σx2= 50 |
Batsman B
|
X |
(X) |
n2 |
|
50 |
-12 |
144 |
|
70 |
+8 |
64 |
|
65 |
+3 |
9 |
|
45 |
-17 |
289 |
|
80 |
+18 |
324 |
|
ΣX=310 |
ΣX2= 830 |
To check the quality of two brands of lightbulbs, their life in burning hours was estimated as under for 100 bulbs of each brand.
|
Life (in hrs) |
No. of bulbs |
||
|
Brand A |
Brand B |
||
|
0–50 |
15 |
2 |
|
|
50–100 |
20 |
8 |
|
|
100–150 |
18 |
60 |
|
|
150–200 |
25 |
25 |
|
|
200–250 |
22 |
5 |
|
|
100 |
100 |
||
(i) Which brand gives higher life?
(ii) Which brand is more dependable?Brand A of light bulbs
|
Life |
No. of |
Mid-points |
d |
d1 |
fd' |
fd2 |
|
|
(in hrs) |
Bulbs (f) |
(m) |
(m – 125) |
|
|||
|
0–50 |
15 |
25 |
–100 |
–2 |
–30 |
60 |
|
|
50–100 |
20 |
75 |
–50 |
–1 |
–20 |
20 |
|
|
100–150 |
18 |
125 |
0 |
0 |
0 |
0 |
|
|
150–200 |
25 |
175 |
50 |
+1 |
25 |
25 |
|
|
200–250 |
22 |
250 |
100 |
2 |
44 |
88 |
|
|
N = 100 |
Σfd' = 19 |
Σfd2 =193 |
|||||
Brand B
|
Life |
No. of |
M.V. |
d |
|
fd' |
fd'2 |
|
|
(in hrs) |
Bulbs |
(m) |
d1 |
||||
|
0–50 |
2 |
25 |
–100 |
–2 |
–4 |
8 |
|
|
50–100 |
8 |
75 |
–50 |
–1 |
–8 |
8 |
|
|
100–150 |
60 |
125 |
0 |
0 |
0 |
0 |
|
|
150–200 |
25 |
175 |
+50 |
+1 |
+25 |
25 |
|
|
200–250 |
5 |
225 |
+100 |
+2 |
+10 |
20 |
|
|
N= 100 |
Σfd' =23 |
Σfd'2 = 61 |
|||||
(i) Since the average life of bulbs of Brand B (136.5) is greater than that of Brand A (134.5 hrs), therefore the bulbs of Brand B givens a higher life.
(ii) Since CV of bulbs of Brand B (27.34%) is less than that of Brand A (51.15%), therefore, the bulbs of Brand B are more dependable.
Average daily wage of 50 workers of a factory was Rs. 200 with a standard deviation of Rs. 40. Each worker is given a raise of Rs. 20. What is the new average daily wage and standard deviation? Have the wages become more or less uniform?
Total increase in wages = 50 × 20 = Rs. 1000
Total of wages before increase worker in wages = 50 × 200 = Rs. 10,000
Total wages after increase in wages
Hence, mean wages will be affected but standard deviation will not be affected as the standard deviation is independent of origin. Have the wages become or less uniform? In order to calculate uniformity wages, we will have to calculate co-efficient of variation.
Afterwards
Now more uniformity in wages has taken place.
In the previous question, calculate the relative measures of variation and indicate the value, which in your opinion is more reliable.
Co-efficient Of range is the relative measure of range. Hence we will calculate coefficient of range.
Co-efficient of Range of Wheat
Co-efficient of Range of Rice
In the same way, we will calculate co-efficient of quartile deviation and co-efficient of variation of both the crops.
Relative measure of variation is more reliable.
If in the previous question, each worker is given a hike of 10% in wages, how are the mean and standard deviation values affected?
With the hike of 10% in wages, the mean will be Rs. 220 / (200 + 20)
There will be affected on standard deviation
Calculate the mean deviation about mean and standard deviation for the following distribution:
|
Classes |
Frequencies |
|
20–40 |
3 |
|
40–80 |
6 |
|
80–100 |
20 |
|
100–120 |
12 |
|
120–140 |
9 |
|
50 |
Calculation of mean deviation about mean:
|
Classes |
f |
mid value (m) |
fx |
|
fd |
|
20–40 |
3 |
30 |
90 |
64.8 |
194.4 |
|
40–80 |
6 |
60 |
360 |
34.8 |
208.8 |
|
80–100 |
20 |
90 |
1800 |
4.8 |
96.0 |
|
100–120 |
12 |
110 |
1320 |
15.2 |
182.4 |
|
120–140 |
9 |
130 |
1170 |
35.2 |
316.8 |
|
N = 50 |
Σfx = 4740 |
Σfd = 998.4 |
Calculation of standard deviation from mean:
|
Classes |
f |
m |
m–90 |
|
fd' |
fd'2 |
|
(d) |
(d') |
|||||
|
20–40 |
3 |
30 |
–60 |
–6 |
–18 |
108 |
|
40–80 |
6 |
60 |
–30 |
–3 |
–18 |
54 |
|
80–100 |
20 |
90 |
0 |
0 |
0 |
0 |
|
100–120 |
12 |
110 |
+20 |
+2 |
24 |
48 |
|
120–140 |
9 |
130 |
+40 |
+4 |
36 |
144 |
|
N = 50 |
Σfd' = 24. |
Σfd'2 = 354 |
A measure of dispersion is a good .supplement to the central value in understanding a frequency distribution. Comment
A measure of dispersion : A good supplement to the central value : A central value condenses the series into a single figure. The measure of central tendencies indicate the central tendency of a frequency distribution in the form of an average. These averages tell us something about the general level of the magnitude of the distribution, but they fail to show anything further about the distribution. The averages represent the series as a whole. One may now be keen to know how far the various values of the series tend to dispense from each other or from their averages. This brings us to yet another important brand of statistical methods, viz. measures of dispersion. Only when we study dispersion alongwith average of series that we can have a comprehensive information about the nature and composition of a statistical series.
In a country, the average income or wealth may be equal. Yet there may be great disparity in its distribution. As a result, thereof, a majority of people may be below poverty line. There is need to measure variation in dispersion and express it as a single figure. It can be further explained with an example. Below are given the family’s incomes of Ram, Rahim and Maria. Ram, Rahim and Maria have four, six and five members in their families respectively.
Family Income
|
St. No. |
Ram |
Rahim |
Maria |
|
1. |
12,000 |
7,000 |
— |
|
2. |
14,000 |
10,000 |
7,000 |
|
3. |
16,000 |
14,000 |
8,000 |
|
4. |
18,000 |
17,000 |
10,000 |
|
5. |
— |
20,000 |
50,000 |
|
6. |
— |
22,000 |
— |
|
Total |
60,000 |
90,000 |
75,000 |
From the table we come to know that each family have average income of Rs. 15,000
considerable differences in individual methods. It is quite obvious that averages try to tell only one aspect of a distribution i.e. representative size of the values. To understand it better, we need to know the spread of values also. The Ram’s family, differences in incomes are comparatively lower. In Rahim’s family, differences are higher and Maria’s family differences are the highest. Knowledge of only average is sufficient. A measure of dispersion improves the understanding of the distribution series.
Define dispersion.
Dispersion measures the extent to which different items tend to disperse away from the central tendency.
How many methods are there to calculate dipersion?
Following are the methods of absolute and relative measures of dispersion:
(i) Absolute measure:Range, quartile deviation, ‘mean deviation, standard deviation, Lorenz curve.
(ii) Relative measure:Coefficient of range, coefficient of quartile deviation, coefficient of mean deviation, standard deviation, coefficient of variation.
Define range.
Range is the difference between the highest value and lowest value in a series.
Define quartile deviation.
Quartile deviation is half of Inter Quartile Range.
Quartile deviation =
How is coefficient of quartile deviation calculated?
Coefficient of quartile deviation is calculated by using the following formula:
Coefficient of QD
Define mean deviation.
Mean deviation is the arithmetic average of the deviations of all the values taken from some average value (mean, median, mode) of the series, ignoring sign (+ or –) of the deviations.
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Define standard deviation.
Standard deviation is the square root of the arithmetic mean of the squares of deviations of the items from their mean value.
What is Lorenz Curve?
Lorenz Curve is a measure of deviation of actual distribution from the line of equal distribution.
What do you mean by coefficient of variation?
Coefficient of variation is a percentage expression of standard deviation. It is 100 times the coefficient of dispersion based on standard deviation of a statistical series.
What is standard deviation?
Standard deviation is the positive square root of the mean of squarred deviations from mean. S.D. is always calculated on the basis of mean only.
What is variance?
Variance is the square of standard deviation. In equation
Variance = (SD)2
Name the four methods available for the calculation of standard deviation of individual series.
(i) Actual mean method (ii) Assumed mean method (iii) Direct method and step deviation method.
What is dispersion?
The degree to which numerical data tend to spread about an average value is called the variation of dispersion. It is an average of second order.
What is measure of dispersion?
The measure of the deviation of the size of items from an average is called a measure of dispersion.
Name the important measures of dispersion.
Range, quartile deviation, mean deviation and standard deviation are the important measures of dispersion.
Define the range.
The range is defined as the difference between the largest and the smallest value of the variable in the given set of values.
R = L —S.
What is mean deviation or mean absolute deviation?
The arithmatic mean of the absolute deviation is called the mean deviation or mean
absolute deviation. Thus 
is the mean deviation of X about the arithmetic mean.
What is standard deviation?
The positive square root of the variance is called the standard deviation of the given value. In equation
Standard Deviation
Standard deviation is always positive. It is absolute measure.
What is the difference between variance and standard deviation?
The variance is the average squared deviation from mean and standard deviation is the square root of variance.
Write down the unique feature of mean deviation.
Mean deviation is the least when taken about median.
Write down the unique feature of the variance.
The variance is unaffected by the choice of assumed mean.
What is coefficient of variation?
Coefficient of variation is the percentage variation in the mean, the standard being treated as the total variation in the mean.
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What is a Lorenz Curve?
Lorenz Curve is a curve which measures the distribution of wealth and income. Now it is also used for the study of the distribution of profits, wages etc.
How do Range and Quartile deviation measure the dispersion?
Range and quartile deviation measure the dispersion by calculating the spread within which the values lie.
What do mean deviation and standard deviation calculate?
Mean deviation and standard deviation calculate the extent to which the values differ from the average.
Which aspect of distribution is indicated and which is not indicated by the averages?
Averages try to tell only one aspect of a distribution i.e. a representative size of the values. It does not tell us the spread of yalues.
Name the measures based on the spread of values.
Range and quartile deviation based on the spread of values.
Give two limitations of range.
1. Range is unduly affected by extreme values.
2. It is not based on all the values.
Not with standing some limitations. Why is range understand and used frequently?
Range is understand and used frequently because of its simplicity.
What is other name of quartile deviation?
The other name of quartile deviation is semi -inter quartile range.
Why is quartile deviation called semi inter - quartile range?
Quartile deviation is called semi-inter quartile range because it is half of the inter -quartile range.
Calculate the range of the following observations:
20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70
Range = L – S = 70 – 20= 50
Calculate quartile deviation from the information given below:
(i) Q3 = 59 (ii) Q1 = 29
Quartile deviation = 
How is Q1 calculated in individual and discrete series?
In individual and discrete series. Q1 is calculated by adopting the following formula:
In continuous series, how is Q1 calculated?
In continuous series Q1 is calculated by applying formula.
Name the measures of dispersion from average.
Mean deviation and standard deviation are the measures of dispersion from average.
How is standard deviation independent of origin?
Standard deviation is independent of origin as it is not affected by the value of constant from which deviations are calculated. The value of the constant does not figure in the standard deviation formula.
Open ended distributions are those distribution in which either the lower limit of the lowest class or the upper limit of the highest class or both are not specified.
Which type of measure is required for comparing the variability of two or more distribution given in different units of measurement?
Relative measure is required for comparing the variability of two or more distributions given in different units.
alculate coefficient range of following distribution:
|
Maths |
No. of Students |
|
0 – 10 |
4 |
|
10 – 20 |
8 |
|
20 – 30 |
12 |
|
30 – 40 |
13 |
Coefficient of Range =
In a town 25% of person earned more than Rs. 50,000 whereas 75% earned more than Rs. 30,000. Calculate Relative value of dispersion.
Relative value of dispersion (co-efficient) 
Why is it better calculate of M.D. from median than that from mean?
It is better to calculate M.D. from median than that from mean because the sum of the deviations taken from median ignoring ± signs is less than the sum of deviations taken from mean.
Write down any one demerit of mean deviation.
Mean deviation cannot be computed with open end class.
How is mean deviation not well - defined measure?
Mean deviation is not well - defined measure because it is calculated from different averages ( Mean, median and mode) and mean deviation calculated from various averages will not be the same.
What is the difference between Variance and standard deviation?
The variance is the average squared deviation from mean and standard deviation is the square - root of variance.
What is the other name of relative measure of dispersion?
The other name of dispersion is coefficient of dispersion.
Write down the relative measures of standard deviation.
Relative measures of standard deviation are (i) Coefficient of standard deviation and (ii) Coefficient of variation.
Write down any one difference between mean deviation and standard deviation.
In the calculation of mean deviation, signs of deviations (+) or (–) are ignored, but in the calculation of standard deviation, signs are not ignored.
What is coefficient of variation?
Coefficient of variation is the percentage variation is the mean, the standard deviation being considered as the total variation in the mean.
How is coefficient of variation calculated?
Coefficient of variation is calculated by dividing the product of standard deviation or related and hundred by mean of the series.
Write down the formula of calculating coefficient if variation.
Coefficient of variation or C.V.
What does higher value of coefficient variation suggest?
Higher value of coefficient variation suggests greater degree of variability and less degree of stability.
What does lower value of coefficient variation suggest?
Lower value of coefficient variation suggest low degree of variability and higher degree of stability, unfromly, homogeneity and consistency.
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Illustrate the meaning of the term dispersion with examples.
Dispersion is a measure of the variation of the items. According to Prof. C.R. Conn, Dispersion is a measure of the extent to which the individual item vary. The measures of dispersion are required to measures the amount of variation of values about the central values.
Example : Suppose the monthly incomes in rupees of five house holds are as Rs. 4500, 6000, 5500, 3750 and 4700.
The arithmatic mean of income is Rs. 4700. The amount of variation in income is shown by deviations from the central values. In this case the deviation from the arithmean are Rs. 390, 1110, 610, 1140 and 190.
What are the properties of a good measure of dispersion?
Properties of a good measure of dispersion:
1. It should be based on all the observations.
2. It should be readily comprehensible.
3. It should be fairly and easily understood.
4. It should be amendable to further algebric treatment.
5. It should be affected as little as possible by fluctuations in sampling.
Give the absolute and relative measures of dispersion.
|
Absolute Measure |
Relative Measure |
|
1. Range |
1. Coefficient of range. |
|
2. Quartile deviation. |
2. Coefficient of Quartile deviation. |
|
3. Mean deviation |
3. Coefficient of Mean deviation. |
|
4. Standard deviation |
4. Coefficient of Standard deviation |
|
5. Lorenz curve. |
Write down the steps involved in the calculation of mean deviation in case of discrete series.
Steps involved in the calculation of mean deviation:Following steps are involved in the calculation of mean Deviation:
1. Find out the mean/median/mode of a series.
2. Find out the deviation of different items from mean/median/ mode.
3. Add the deviations ignoring positive and negative signs. Treat all deviations as positive.
4. Calculate mean deviation by dividing the sum total of the deviation by the number of items.
Write down the steps involved in the calculation of mean deviation for the discrete series.
Steps : 1. Find out central tendency of the series (mean or median) from which deviations are to be taken.
2. Take deviation of different items in the series from central tendency ignoring signs (+,–). Express it as | dx | or (| dm |).
3. Multiply each deviation value by frequency facing it.
4. Add the multiplies and express it as Σ?(d).
5. Divide Σ?(d) by sum total of frequency. The resultant value will be mean deviation.
Write down the features of mean deviation.
These are features of mean deviation:
1. Mean deviation is rigidly defined.
2. It depends on all the values of the variable.
3. It is based on absolute deviations from central values.
4. It is easy to understand.
5. It involves harder calculation than the range and quartile deviation.
6. It is amendable to algebraic treatment.
7. The units of measurement of the mean deviation are the same as those of the variable.
Differentiate between Mean Deviation and Standard Deviation.
Difference between Mean Deviation and Standard Deviation:
|
Mean Deviation |
Standard Deviation |
|
1. In calculating mean deviation. algebraic signs are ignored. |
1. In calculating standard deviation, algebraic signs are taken into account. |
|
2. Mean or median is used in calculating the mean deviation. |
2. Only mean is used in calculating the standard deviation. |
What are the uses of coefficient of variation?
Coefficient of variation is used to compare the variability, homogeneity, stability and uniformity of two different statistical series. Higher value of coefficient variation suggests greater degree of variation and lesser degree of stability. On the other hand, a lower value of coefficient variation suggests lower degree of variability and higher degree of stability, uniformity, homogeneity and consistency.
Explain merits and demerits of quartile Deviation.
Merits : 1. It is easy understand and to calculate.
2. It is unaffected by the extreme values.
3. It is quite satisfactory when only the middle half of the group is dealt with.
Demerits : 1. It ignores 50 per cent of the extreme items.
2. It is not capable of algebraic treatment.
3. This is not useful when extreme items are to be given special height.
Write down the merits of mean deviation.
Merits of Mean Deviation:
1. It is easy to understand mean Deviation.
2. Mean Deviation is less affected by extreme value than the Range.
3. Mean deviation is based on all the items of the series. It is therefore, more representative than the Range or Quartile Deviation.
4. It is very simple and easy measure of dispersion.
Demerits of Mean Deviation:
1. Mean deviation is not capable of algebraic treatment, because it ignores plus and minus signs.
2. It is not a well-defined measure since mean deviation from different averages (mean, median and mode) will not be the same.
The height of 11 men were 61, 64, 68, 69, 67, 68, 66, 70, 65, 67 and 72 inches. Calculate the range of the shortest man is omitted, what is the percentage change in the range?
1. Range = L = S
= 72 - 61 = 11 inches
2. New Range (Shortest man is omitted)
= L - S
= 72 - 64 = 8
Change in range = 11-8 = 3 inches
percentage change in range = 3/11 ×100 = 27.2%
What will be the effect of change of origin and change of scale on the standard deviation, mean and variance?
Change of origin and change of scale: Following are the effects of change of origin and change of scale on the mean, standard deviation and variance.
1. Any constant added or substracted (change of origin) than the standard deviation of original data and of change data after addition or substraction will not change but the mean of new data will change.
2. Any constant multiplied or divided (Change of scale) then mean, standard deviation and variation will change of the new changed data.
The following table gives you the height of 100 persons. Calculate dispersion by range method.
|
Height in Centimetres |
No. of Persons |
|
Below 162 |
2 |
|
Below 163 |
8 |
|
Below 164 |
19 |
|
Below 165 |
32 |
|
Below 166 |
45 |
|
Below 167 |
58 |
|
Below 168 |
85 |
|
Below 169 |
93 |
|
Below 170 |
100 |
Calculation of dispersion by Range method:
|
Height in Centimetres |
No. of Persons |
|
161–162 |
2 |
|
162–163 |
6 |
|
163–164 |
11 |
|
164–165 |
13 |
|
165–166 |
13 |
|
166–167 |
13 |
|
167–168 |
27 |
|
168–169 |
8 |
|
169–170 |
7 |
|
Total |
100 |
How is dispersion of the series different from average of the series?
Average of the series refers to the central tendency of the series. It represents behaviour of all the items in the series. But different items tend to different from each other and from the averages. Dispersion measures the extent to which different items tend to disperse away from the central tendency.
Why should we measure dispersion about some particular value?
We should measure dispersion about some particular value because in that case (i) We can assess how precise is the central tendency as the representative value of all the observations in the series. Greater value of dispersion implies lesser representativeness of the central tendency and vice versa.
(ii) We can precisely asses how scattered are the actual observation from their representative value.
Why is standard deviation also known as the root mean square deviation?
Standard deviation is also known as the root mean square deviation because it is the square root of the means of the square deviation from the arithmetic mean. In the calculation of standard deviation, first the arithmetic average is calculated, and the variations of various items from arithmetic averages are squared. The squared deviations are totalled and the sum is divided by the number of items. The square root of the resulting figure is the standard deviation of the series. The S.D. is denoted by the Greek letter. σ (Sigma) Symbalically
Give the comparison of alternative measures of dispersion.
Comparison of alternative measures of the dispersion has been discussed below :
1. Rigidly defined:All the four measures-the range, quartile deviation, mean deviation and standard deviation are rigidly defined. There is no vagueness in their definition.
2. Ease of calculation:The range is the easiest one to calculate. Quartile deviation requires calculation of the upper and lower quartiles but that is also easy enough. However, the mean deviation and standard deviation require a little more systematic calcualtion. They, too are easy.
3. Simple interpretation:All measures of dispersion are easy to interpret. While the range and quartile deviation measure dispersion in a general way the mean deviation and standard deviation measure dispersion in terms of deviations from a central value. Thus the mean deviation and standard deviation give a better idea about the dispersion of values within the range.
4. Based on all values:The range and quartile deviation do not depend on all values, whereas, the mean deviation and standard deviation use all values of the variable. The range is affected the most by extreme values.
5. Amendable to algebraic treatment:The standard deviation is perhaps the easiest for analytical work. Other measures can be also dealt with analytically but derivation are harder.
What are the four alternative measures of absolute dispersion? Discuss their properties.
The four alternative measure of absolute dispersion are :
(i) Range, (ii) Quartile Deviation, (iii) Mean deviation, (iv) Standard Deviation.
1. Features of the range:
(a) It is rigidly defined.
(b) It is easy to calculate and simple to interpret.
(c) It does not depend on all values of the variables.
(d) It is unduly affected by extreme values.
(e) The range depends on the units of measurement of the variable.
2. Features of Quartile deviation:
(a) It is rigidly defined.
(b) It is easy to calculate and simple to interpret.
(c) It does not depend on all values of the variable.
(d) The units of measurement of the quartile deviation are the same as those of the variable.
3. Features of mean deviation:
(a) It is rigidly defined.
(b) It depends on all values of the variable.
(c) It is based on absolute deviations from a central values.
(d) It is easy to understand.
(e) It involves harder calculations than the range and quartile deviation.
(f) It is amendable to algebraic treatment.
(g) The units of measurement of the mean deviation are the same as those of the variable.
4. Features of Standard deviation:
(a) It is the best measure of the variation because it is based every item of the series and further algebraic treatment is possible.
(b) It is not very much affected by fluctuation of sampling.
(c) It is the only measure for calculating combined standard deviation of two or more graphs.
(d) It is a definite measure of dispersion.
“ The coefficient of variation is a relative measure of disperison’. We may calculate coefficient of variation using any of the measure of dispersion such as range, quartile deviation, mean deviation and standard deviation.
Illustrate the use of coefficient of variation in these cases.
There are two types of dispersion absolute measure and relative measures of dispersion. Absolute measures of dispersion are measured in the same units as those of variables considered. This feature of measures of dispersion may create difficulty if we want to compare dispersion in two sets of values which have (i) different central values and (ii) different units of measurement.
In order to overcome this difficulty it is desirable to eliminate the units. This can be done if we use a relative measure of dispersion which is a pure number and do not depend on units of measurement. The relative measure of dispersion is called the cofficient of variation. It may be expressed as ratio or express it in percentage.
The most commonly used coefficient of variation is ratio of standard deviation we may also express in percentage as
Where σ is the standard deviation and m is arithmetic mean.
We may also compute the coefficient of variation as
if we are using the range as measure of. dispersion.
if we are using quartile deviation as measure of dispersion.
Similarly using mean deviation,
Give the formulae of range, Quartile deviation, mean deviation, standard deviation (Absolute and relative measures both.)
1. Range
Range = L – S
L = Largest item
S = Smallest item
2. Quartile deviation
Q1 = Lower Quartile.
Q3 = Upper Quartile.
3. Mean Deviation
Individual Observation:
4. Standard Deviation
Individual observators
Actual Mean Method
An analysis of the weekly wages paid to workers in two firms A and B give the following result:
|
Firm A |
Firm B |
|
|
No. of Workers |
586 |
648 |
|
Average weekly wages |
Rs. 52.5 |
Rs. 47.5 |
|
Variance of the distribution of wages |
100 |
121.0 |
Which firm “A” or “B” has greater variability in individual wage.
Firm A
Since coefficient of variation is higher in case of firm B, it shows greater variability in individual wages.
Prove by an example that the variance is unaffected by the choice of the assumed mean.
We take the following example for proving that the variance is uneffected by the choice of the assumed mean.
Example:Calculate variance of 25, 50, 45, 30, 70, 42, 36, 48, 34 and 60 by actual mean assumed mean method.
(a) Calculation of Variance by Actual Mean Method
|
Values of X |
|
x2 |
|
25 |
–19 |
361 |
|
50 |
+6 |
36 |
|
45 |
+1 |
1 |
|
30 |
–14 |
196 |
|
70 |
–26 |
676 |
|
42 |
–2 |
4 |
|
36 |
–8 |
64 |
|
48 |
+4 |
16 |
|
34 |
–10 |
100 |
|
60 |
+16 |
256 |
|
ΣX = 440 |
ΣX2 = 1710 |
(b) Calculation of Variance by Assumed Mean Method
|
Values X |
(X–45) d |
d2 |
|
25 |
–20 |
400 |
|
50 |
+5 |
25 |
|
45 |
0 |
0 |
|
30 |
–15 |
225 |
|
70 |
+25 |
625 |
|
42 |
–3 |
9 |
|
36 |
–9 |
81 |
|
48 |
+3 |
9 |
|
34 |
–11 |
121 |
|
60 |
+15 |
225 |
|
N=10 |
Σd = –10 |
Σd2 =1720 |
The distribution of the cost of production (in rupees) of a quintal of wheat in 50 farms is as follows:
|
Cost (in Rs.) |
Number of Farms |
|
40–50 |
3 |
|
50–60 |
6 |
|
60–70 |
12 |
|
70–80 |
18 |
|
80–90 |
9 |
|
90–100 |
2 |
|
Total |
50 |
(a) Calculate the variance.
(i) by direct method.
(ii) by step deviation method and compare your results with the mean deviation about the arithmatic mean.
(b) Calculate the coefficient of variation by using
(i) the standard deviation of costs and
(ii) the mean deviation of cost about the arithmatic mean and compare the two. What is your conclusion about variation of cost.
Calcualtion of Variance by Direct Method.
|
Class Interval |
f |
Mid Point |
fx |
|
fd |
d2 |
fd2 |
|
40-50 |
3 |
45 |
135 |
-26 |
78 |
676 |
2028 |
|
50-60 |
6 |
55 |
330 |
-16 |
96 |
256 |
1536 |
|
60-70 |
12 |
65 |
780 |
-6 |
72 |
36 |
432 |
|
70-80 |
18 |
75 |
1350 |
4 |
72 |
16 |
288 |
|
80-90 |
9 |
85 |
765 |
14 |
126 |
196 |
1764 |
|
90-100 |
2 |
95 |
190 |
24 |
48 |
576 |
1152 |
|
Σf = 50 |
Σf = 3550 |
Σfd2 = 7200 |
Calculation of Variance by Step Deviation
|
Class |
? |
Mid |
d |
||||
|
Interval |
Point (X) |
d’ |
d’2 |
?d' |
?d'2 |
||
|
40-50 |
3 |
45 |
–30 |
–3 |
9 |
–9 |
27 |
|
50-60 |
6 |
55 |
–20 |
–2 |
4 |
–12–33 |
24 |
|
60-70 |
12 |
65 |
–10 |
–1 |
1 |
–33 |
–12 |
|
70-80 |
18 |
75 |
0 |
0 |
0 |
0 |
0 |
|
80-90 |
9 |
85 |
10 |
+1 |
1 |
9+13 |
9 |
|
90-100 |
2 |
95 |
20 |
+2 |
4 |
4 |
8 |
|
Σ? = 80 |
Σd = 90 |
Σfd' = 20 |
Σfd'2 = 80 |
(b) Calculation of coefficient of variation:
1. Variance coefficient (From S.D.)
Briefly explain the various measures calculated from standard deviation.
Measures calculated from standard deviation :
Mainly following measures are calculated from standard deviation :
1. Coefficient of standard deviation : It is a relative measure of standard deviation. It is calculated to compare the variability in two or more than two series. It is calculated by dividing the standard deviation by arithmetic mean of data symbolically.
2. Coefficient of Variance : It is most propularly used to measure relative variation of two or more than two series. It shows the relationship between the S.D. and the arithmetic mean expressed in terms of percentage. It is used to compare uniformly, consistency and variability in two different series.
3. Variance : It is the square of standard deviation. It is closely related to standard deviation. It is the average squared deviation from mean where as standard deviation is the square is the square root of variance. Symbolically
A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are :
|
X |
Y |
|
25 |
50 |
|
85 |
70 |
|
40 |
65 |
|
80 |
45 |
|
120 |
80 |
(a) Calculate coefficient of standard deviation, variance and coefficient of variation.
(b) Which batsman should be selected if we wants.
(i) a higher run getter, or
(ii) a more reliable batsman in the team ?
a)
|
Scores |
Batsman x |
|
|
|
|
|
|
25 |
–45 |
2025 |
|
85 |
+15 |
225 |
|
40 |
–30 |
900 |
|
80 |
+10 |
100 |
|
120 |
+50 |
2500 |
|
ΣX=350 |
Σx2= 570 |
|
Batsman Y
|
Scores |
|
|
|
50 |
–12 |
144 |
|
70 |
8 |
64 |
|
65 |
3 |
9 |
|
45 |
–17 |
289 |
|
80 |
18 |
324 |
|
Σx=310 |
Σx2= 830 |
(i) Batsman X should be selected as a higher run getter as his average score (70 runs) is greater than that of Y (i.e. 62 runs)
(ii)Batsman Y is a more reliable batsman in the team because his coefficient of variance (20.77) is less than that of batsman X (c.v. 48.44)
Calculate the standard deviation of the following values by following methods:
(i) Actual Mean Method, (ii) Assumed Mean Method, (iii) Direct Method, (iv) Step Deviation Method.
5, 10, 25, 30, 50.
(i) Calculation of Standard Deviation by Actual Mean Method:
|
X |
d | d2 |
5 | –19 | 361 |
10 | –14 | 196 |
25 | +1 | 1 |
30 | +6 | 36 |
50 | +26 | 676 |
ΣX = 120 | 0 | Σd2 = 1270 |
(ii) Calculation of Standard Deviation by Assumed Mean Method:
X | d | d2 |
5 | –20 | 400 |
10 | –15 | 225 |
25 | 0 | 0 |
30 | +5 | 25 |
50 | +25 | 625 |
–5 | 1275 |
(iii) Calculation of Standard Deviation by Direct Method : Standard Deviation can also be calculated from the values directly, i.e., without taking deviations, as shown below:
X | x2 |
5 | 25 |
10 | 100 |
25 | 625 |
30 | 900 |
|
50 |
2500 |
|
ΣX = 120 |
ΣX2 = 4150 |
(iv) Calculation of Standard Deviation by Step Deviation Method : The values are divisible by a common factor, they can be so divided and standard deviation can be calculated from the resultant values as follows :
Since all the five values are divisible by a common factor 5, we divide and get the following values:
|
x |
x2 |
d |
d2 |
|
5 |
1 |
–3.8 |
14.44 |
|
10 |
2 |
–2.8 |
7.84 |
|
25 |
5 |
+0.2 |
0.04 |
|
30 |
6 |
+1.2 |
1.44 |
|
50 |
10 |
+5.2 |
27.04 |
|
N = 5 |
0 |
50.80 |
Alternative Method:Alternatively, instead of dividing the values by a common factor, the deviations can be divided by a common factor. Standard Deviation can be calculated as shown below:
|
x |
d |
d' |
d2 |
|
5 |
–20 |
–4 |
16 |
|
10 |
–15 |
–3 |
9 |
|
25 |
0 |
0 |
0 |
|
30 |
+5 |
+1 |
1 |
|
50 |
+25 |
+5 |
25 |
|
N = 5 |
–1 |
51 |
Deviations have been calculated from an arbitrary value 25. Common factor of 5 has been used to divide deviations.
Calculate Mean Deviation from the following table using :
(i) Actual Mean Method
(ii) Assumed Mean Method
(iii) Step Deviation Method
|
Profits of Companies (Rs. in lakhs) |
Number of |
|
Class-intervals |
Companies frequencies |
|
10 – 20 |
5 |
|
20 – 30 |
8 |
|
30 – 50 |
16 |
|
50 – 70 |
8 |
|
70 – 80 |
3 |
|
40 |
(i) Calculation of S.D. with the help of Actual Mean Method :
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
|
CI |
f |
m |
fm |
d |
fd |
fd2 |
|
10–20 |
5 |
15 |
75 |
–25.5 |
–127.5 |
3251.25 |
|
20–30 |
8 |
25 |
200 |
–15.5 |
–124.0 |
1922.00 |
|
30–50 |
16 |
40 |
640 |
–0.5 |
8.0 |
4.00 |
|
50–70 |
8 |
60 |
480 |
+19.5 |
+156.0 |
3042.00 |
|
70–80 |
3 |
75 |
225 |
+34.5 |
+103.5 |
3570.75 |
|
Σf=40 |
Σfm=1620 |
Σfd=0 |
Σfd2= 11790.00 |
(ii) Calculation of Standard Deviation by Assumed Mean Method :
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
|
CI |
f |
m |
d |
fd |
fd2 |
|
10–20 |
5 |
15 |
–25 |
–125 |
3125 |
|
20–30 |
8 |
25 |
–15 |
–120 |
1800 |
|
30–50 |
16 |
40 |
0 |
0 |
0 |
|
50–70 |
8 |
60 |
+20 |
160 |
3200 |
|
70–80 |
3 |
75 |
+35 |
105 |
3675 |
|
Σf 40 |
Σfd=+20 |
Σfd2=11800 |
(iii) Calculation of Standard Deviation by Step Deviation Method :
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
|
CI |
f |
m |
d |
d' |
fd' |
fd'2 |
|
10–20 |
5 |
15 |
–25 |
–5 |
–25 |
125 |
|
20–30 |
8 |
25 |
–15 |
–3 |
-24 |
72 |
|
30–50 |
16 |
40 |
0 |
0 |
0 |
0 |
|
50-70 |
8 |
60 |
+20 |
+4 |
+32 |
128 |
|
70–80 |
3 |
75 |
+35 |
+7 |
+21 |
147 |
|
40 |
+4 |
472 |
Find the Standard Deviation of the height of 100 students:
|
Height in inches |
Frequency |
|
Less than 62.5 |
5 |
|
Less than 65.5 |
23 |
|
Less than 68.5 |
65 |
|
Less than 71.5 |
92 |
|
Less than 74.5 |
100 |
First convert cumulative frequency with class interval.
Calculation of Standard Deviation
|
Height in Inches |
Frequency |
M.P. |
M–67 |
|
fd' |
fd'2 |
|
X |
(f) |
(m) |
d |
d' |
||
|
59.5-62.5 |
5 – 0 = 5 |
61 |
–6 |
–2 |
–10 |
20 |
|
62.5 – 65.5 |
23–5=18 |
64 |
–3 |
–1 |
–18 |
18 |
|
65.5 – 68.5 |
65–23 = 42 |
67 |
0 |
0 |
0 |
0 |
|
68.5 – 71.5 |
92–65 = 27 |
70 |
+3 |
+1 |
+27 |
27 |
|
71.5 – 74.5 |
100–92 = 8 |
73 |
+6 |
+2 |
+16 |
32 |
|
N = 100 |
Σfd'=15 |
Σfd'2 = 97 |
Calculate mean standard deviation and mean deviation about mean from the following distribution :
|
Marks |
Students |
|
More than 20 |
50 |
|
More than 40 |
47 |
|
More than 80 |
41 |
|
More than 100 |
21 |
|
More than 120 |
9 |
Convert cumulative frequency with interval.
Calculation of Mean and Standard Deviation
|
Marks |
Frequency |
Mid-points |
m–90 d |
|
fd' |
fd'2 |
|
(x) |
(f) |
(m) |
d |
d' |
||
|
20 – 40 |
50 – 47 = 3 |
30 |
–60 |
–6 |
–18 |
108 |
|
40 – 80 |
47 – 41 =6 |
60 |
–30 |
–3 |
–18 |
54 |
|
80 – 100 |
41 – 21 = 20 |
90 |
0 |
0 |
0 |
0 |
|
100 – 120 |
21 – 9 = 12 |
110 |
+20 |
+2 |
24 |
48 |
|
120 – 140 |
9 – 0 = 9 |
130 |
+40 |
+4 |
36 |
144 |
|
N = 50 |
Σfd'=24 |
Σfd'2 = 354 |
Calculation of Mean Deviation from mean
|
Marks |
Frequency |
Mid-points |
m–94.8 |
f|D| |
|
X |
(f) |
(m) |
||
|
20 – 40 |
3 |
30 |
64.8 |
194.4 |
|
40 – 80 |
6 |
60 |
34.8 |
208.8 |
|
80 – 100 |
20 |
90 |
4.8 |
96 |
|
100 – 120 |
12 |
110 |
15.2 |
182.4 |
|
120 – 140 |
9 |
130 |
15.2 |
316.8 |
|
N = 50 |
Σf|D| = 998.4 |
Calculate the arithmetic mean and standard deviation of the following values:
(i) without grouping
(ii) grouping the value in classes 140–145, 145–150 .............
(iii) grouping them in classes 140–150, 150–160 ............
|
140 |
143 |
143 |
146 |
146 |
|
146 |
154 |
156 |
159 |
162 |
|
164 |
174 |
166 |
166 |
167 |
|
167 |
168 |
168 |
169 |
169 |
|
169 |
171 |
175 |
175 |
176 |
|
176 |
178 |
180 |
182 |
182 |
|
182 |
182 |
182 |
183 |
184 |
|
186 |
188 |
190 |
190 |
191 |
|
191 |
192 |
195 |
202 |
217 |
(i) Without grouping 
|
X |
d |
d2 |
|
140 |
–35 |
1225 |
|
143 |
–32 |
1024 |
|
143 |
–32 |
1024 |
|
146 |
–29 |
841 |
|
146 |
–29 |
841 |
|
146 |
–29 |
841 |
|
154 |
–21 |
441 |
|
156 |
–19 |
361 |
|
159 |
–16 |
256 |
|
162 |
–13 |
169 |
|
164 |
–11 |
121 |
|
164 |
–11 |
121 |
|
166 |
–9 |
81 |
|
166 |
–9 |
81 |
|
167 |
–8 |
64 |
|
167 |
–8 |
64 |
|
168 |
–7 |
49 |
|
168 |
–7 |
49 |
|
169 |
–6 |
36 |
|
169 |
–6 |
36 |
|
169 |
–6 |
36 |
|
171 |
–4 |
16 |
|
175 |
0 |
0 |
|
175 |
0 |
0 |
|
176 |
1 |
1 |
|
176 |
1 |
1 |
|
178 |
3 |
9 |
|
180 |
5 |
25 |
|
182 |
7 |
49 |
|
182 |
7 |
49 |
|
182 |
7 |
49 |
|
182 |
7 |
49 |
|
182 |
7 |
49 |
|
183 |
8 |
64 |
|
184 |
9 |
81 |
|
186 |
11 |
121 |
|
188 |
13 |
169 |
|
190 |
15 |
225 |
|
190 |
15 |
225 |
|
191 |
16 |
256 |
|
191 |
16 |
256 |
|
192 |
17 |
289 |
|
195 |
20 |
400 |
|
202 |
27 |
729 |
|
227 |
52 |
2704 |
|
13577 |
(ii) Grouping the values in classes 140-145,145-150
(iii) Grouping them in classes 140–150, 150–160.
|
Classes |
Tally |
Frequency |
M |
d' |
fd' |
d |
fd'2 |
|
140–150 |
|
6 |
145 |
–4 |
–24 |
16 |
96 |
|
150–160 |
|
3 |
155 |
–3 |
–9 |
9 |
27 |
|
160–170 |
|
12 |
165 |
–2 |
–24 |
4 |
48 |
|
170–180 |
|
6 |
175 |
–1 |
–6 |
1 |
6 |
|
180–190 |
|
10 |
185 |
0 |
0 |
0 |
0 |
|
190-200 |
|
6 |
195 |
1 |
6 |
1 |
6 |
|
200-210 |
|
1 |
205 |
2 |
2 |
4 |
4 |
|
210–220 |
|
0 |
215 |
3 |
0 |
9 |
0 |
|
220–230 |
|
1 |
225 |
4 |
4 |
16 |
16 |
|
Σf = 45 |
Σfd'=–51 |
Σfd' = 203 |
The Standard Deviation of height measured in inches will be larger than the Standard Deviation of the height measured in ft. for the same group of individuals. Comment on the validity or otherwise of the statement with appropriate illustration.
The statement is totally valid. The least is that Standard Deviation is an absolute measure. When the units of measurement are different the less the measurement will, the more will the Standard Deviation. With the increase in the measurement unit, the Standard Deviation will decrease. It is clear from the following illustrations.
Suppose we are given the heights of 5 persons in feet. such as 2, 4, 6, 8, 10. With the help of the data we will calculate the S.D.
|
Height in feet |
D |
D2 |
|
(X) |
|
|
|
2 |
–4 |
16 |
|
4 |
–2 |
4 |
|
6 |
0 |
0 |
|
8 |
2 |
4 |
|
10 |
4 |
16 |
|
ΣX = 30 |
ΣD2 = 40 |
Now we calculate S.D. taking the heights of 5 same persons in inches.
|
Height in inches |
D |
D2 |
|
(X) |
|
|
|
24 |
–48 |
2304 |
|
48 |
–24 |
576 |
|
72 |
0 |
0 |
|
96 |
+24 |
576 |
|
120 |
+48 |
|
|
ΣX = 360 |
ΣD2 = 5760 |
Thus, we see that S.D. has increased 12 times.
Using median and arithmetic mean respectively. Calculate mean deviation and coefficient of mean deviation from the following data:
|
Size of items |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
13 |
|
Frequency |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
(i) Calculation of M.D. from Arithmetic Mean
|
Size X |
Frequency f |
fx |
Deviation from Mean (9.83) dx |
Product of Frequency and |–dx| (f|–dx|) |
|
5 |
4 |
20 |
4.83 |
19.32 |
|
6 |
5 |
30 |
3.83 |
19.15 |
|
7 |
6 |
42 |
2.83 |
16.98 |
|
8 |
7 |
56 |
1.83 |
12.81 |
|
9 |
8 |
72 |
0.83 |
6.64 |
|
10 |
9 |
90 |
0.17 |
1.53 |
|
11 |
10 |
110 |
1.17 |
11.70 |
|
12 |
11 |
132 |
2:17 |
23.87 |
|
13 |
12 |
156 |
3.17 |
38.04 |
|
Σf =72 |
Σfx = 708 |
Σf |–dx| = 150.04 |
(ii) Calculation of M.D. from Median
|
Size of Items (C) |
Frequency (f) |
Cumulative Frequency (cf) |
Deviation from Median M = 10 |dm | = (X – M) |
f |dm | |
|
5 |
4 |
4 |
5 |
20 |
|
6 |
5 |
9 |
4 |
20 |
|
7 |
6 |
15 |
3 |
18 |
|
8 |
7 |
22 |
2 |
14 |
|
9 |
8 |
30 |
1 |
8 |
|
10 |
9 |
39 |
0 |
0 |
|
11 |
10 |
49 |
1 |
10 |
|
12 |
11 |
60 |
2 |
22 |
|
13 |
12 |
72 |
3 |
36 |
|
N = 72 |
ΣX |dm| = 148 |
The distribution of the cost of production (in rupees) of a quintal of wheat in 50 farms in as follows:
|
Cost (in rupees) |
40–50 |
50–60 |
60–70 |
70–80 |
80–90 |
90–100 |
Total |
|
Number of farms |
3 |
6 |
12 |
18 |
9 |
2 |
50 |
(a) Calculate the variance
(i) by direct method, (ii) by step deviation method and compare your results with the mean deviation about the arithmetic mean.
(b) Calculate the coefficient of variation by using:
(i) The standard devitation of cost and (ii) the mean deviation of costs about the arithmetic mean, and compare the two. What is your conclusion about variation of cost?
(a) (i) Calculation of Variance by Direct Method
|
Class |
Frequency |
Mid-points (X) |
(xf) |
|
f|D| df |
fm |
d2 |
fd2 |
|
40 – 50 |
3 |
45 |
135 |
–26 |
78 |
135 |
676 |
2028 |
|
50 – 60 |
6 |
55 |
330 |
–16 |
96 |
330 |
256 |
1536 |
|
60 – 70 |
12 |
65 |
780 |
–6 |
72 |
780 |
36 |
432 |
|
70 – 80 |
18 |
75 |
1350 |
4 |
72 |
1350 |
16 |
288 |
|
80 – 90 |
9 |
85 |
765 |
14 |
126 |
765 |
196 |
1764 |
|
90 – 100 |
2 |
95 |
190 |
24 |
48 |
190 |
576 |
1152 |
|
50 |
Σnf = 3550 |
Σd = 90 |
Σ × P = 492 |
Σfd2 = 7200 |
(ii) By Step Deviation Method :
|
Class |
f |
m |
d(M–n) |
d'2 |
fd' |
fd'2 |
|
40 – 50 |
3 |
45 |
–3 |
9 |
–9 |
27 |
|
50 – 60 |
6 |
55 |
–2 |
4 |
–12 |
24 |
|
60 – 70 |
12 |
65 |
–1 |
1 |
–12 |
12 |
|
70 – 80 |
18 |
75 |
0 |
0 |
0 |
0 |
|
80 – 90 |
9 |
85 |
1 |
1 |
9 |
9 |
|
90 – 100 |
2 |
95 |
2 |
4 |
4 |
8 |
|
Σfd'=-20 |
Σfd'2= 80 |
(b) Calculation of Coefficient of Variation
A study of certain examination results of 1000 students at the year 2000 gave average marks secured as 50% with a standard deviation of 3%. A similar study of 2001 revealed average marks secured and standard deviation as 55% and 5% respectively. Have the results improved?
Results have not improved in 2001 as the relative dispersion in 2001 is more than that of in 2000.
Prove with an example that Q. D. is the average difference of the quartiles from Median.
In order to prove that Q. D. is the average difference of the quartiles from median. We calculate Q1, Q3, Q.D and median from the following data :
20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70.
Hence, it is proved Q.D. (here 11) is the average difference of the quartiles from the median.
Prove that mean deviation calculated about mean will be greater than that calculate about median.
In order to prove the statement given in the question we calculate mean deviation about mean and mean deviation from median and compare what is greater :
2, 4, 7, 8 and 9.
Mean deviation about Mean
|
X |
D |
|
2 |
4 |
|
4 |
2 |
|
7 |
1 |
|
8 |
2 |
|
9 |
3 |
|
ΣX = 30 |
ΣD = 12 |
|
X |
D (X–7) |
|
2 |
5 |
|
4 |
3 |
|
7 |
0 |
|
8 |
1 |
|
9 |
2 |
|
N = 5 |
ID = 11 |
Mean deviation about mean is 2.4. and mean deviation about median is 2.2. Hence, proved that mean deviation about mean is greater than mean deivation about median.
Prove that mean deviation is based on all values. A change in even one value will effect of.
In order to prove the statement given in the question, we calculate the mean deviation of the following data: 2, 4, 7, 8, 9.
|
X |
D |
|
2 |
4 |
|
4 |
2 |
|
7 |
1 |
|
8 |
2 |
|
9 |
3 |
|
ΣX = 30 |
ΣD = 12 |
Now change one value i.e. we take 14 in place of 9.
|
X |
D |
|
2 |
5 |
|
4 |
3 |
|
7 |
0 |
|
8 |
1 |
|
14 |
7 |
|
ΣX = 35 |
ΣD = 16 |
In this way we see that mean deviation changes with a change in even one value. Earlier, the mean deviation was 2.4. After the change in one value it is 3.2.
The Standard deviation of height measured in inches will be larger than the Standard Deviation of heights measured in foot for the same group of individuals. Comment on the validity or otherwise of this statement with appropriate explanation.
The statement given in the question is absolutely correct. The reason is that standard deviation is an absolute measure. It can create problem when units of measurement are different. The lesser the measurement the higher the standard deviation and vice-versa. If we measure the height in inches the instead of foot, than the S.D. will increase 12 times. It has been explained below with an example.
Suppose we are given the height of 5 good in foot in the following table. With the help of the table, we will calculate S.D.
|
Height in foot (X) |
D |
D2 |
|
2 |
–4 |
16 |
|
4 |
–2 |
4 |
|
6 |
0 |
0 |
|
8 |
2 |
4 |
|
10 |
4 |
16 |
|
ΣX = 30 |
ΣD2 = 40 |
Now we will calculate S.D. taking the height of 5 same persons in inches.
|
Height in inches (X) |
D |
D2 |
|
24 |
–48 |
2304 |
|
48 |
–24 |
576 |
|
72 |
0 |
0 |
|
96 |
+24 |
576 |
|
120 |
+48 |
2304 |
|
ΣX = 360 |
ΣD2 = 5760 |
Hence, it is proved that the deviation of height measured in inches will be larger than heights measured in foot.
Sponsor Area
From the following table calculate
1/7 mean deviation from mean.
|
Profits of Companies (Rs. in lakhs) |
No. of Companies |
|
10–20 |
5 |
|
20–30 |
8 |
|
30–50 |
16 |
|
50–70 |
8 |
|
70–80 |
3 |
|
40 |
Calculation from mean:
|
C.I. |
f |
m.p. |
fm. |
|d| |
f|d| |
|
10–20 |
5 |
15 |
75 |
25.5 |
127.5 |
|
20–30 |
8 |
25 |
200 |
15.5 |
124.0 |
|
30–50 |
16 |
40 |
640 |
0.5 |
8.0 |
|
50–70 |
8 |
60 |
480 . |
19.5 |
156.0 |
|
70–80 |
3 |
75 |
225 |
34.5 |
103.5 |
|
Σf = 40 |
Σfm = 1620 |
Σf|d| = 519.0 |
How is dispersion of the series different from the average of the series? What will be the effect of change of origin and change of scale on S.D. mean and variance series?
a) Difference between dispersion of series and average of series : Averages of series in known as the measures of central tendency. An average indicates respresentative value of the series around which other value of the series tend to converage. So the average represents the series as a whole. In the other hand dispersion is the measure of the variationes of the items. It helps us in knowing about the composition of a series or the dispersal of values on the either side of the central tendency.
(b) Effect of change of origin and change of scale in the S.D. mean and variance : Change of origin i.e. any constant added or subtracted will have no effect on standard deviation but it will change the mean.
On the other hand change of scale (any constant multiplied or divided) will change the mean, standard deviation and variance.
The heights of 11 men are 61, 64, 68, 69, 67, 68, 66, 70, 65, 67 and 72 inches. Calculate range of the man of the least height in removed. What will be the percentage change in the range?
1. Range =L – S
= 72 – 61 = 11 inches.
2. New range (after removing the man of least height) = 72 – 64 = 8 inches.
3. Change in range = 11 – 8 = 3 inches.
Percentage change in Range
Below is given the height of 100 men. Calculate dispersion by range method.
|
Height in Centimetres |
No. of Persons |
|
Less than 162 |
2 |
|
Less than 163 |
8 |
|
Less than 164 |
19 |
|
Less than 165 |
32 |
|
Less than 166 |
45 |
|
Less than 166 |
58 |
|
Less than 167 |
85 |
|
Less than 168 |
93 |
|
Less than 169 |
100 |
Calculation of Dispersion by Range Method:
|
Height in Centimetres |
No. of Persons |
|
161 – 162 |
22 |
|
162 – 163 |
6 |
|
163 – 164 |
11 |
|
164 – 165 |
13 |
|
165 – 166 |
13 |
|
166 – 167 |
13 |
|
167 – 168 |
27 |
|
168 – 169 |
8 |
|
169 – 170 |
7 |
|
Total |
100 |
Calculate the inter quartile range, quartile deviation and coefficient of quartile deviation of the following frequency distribution relating to bonus paid up to workers.
|
Bonus (in Rs.) |
No. of Workers |
|
300–320 |
2 |
|
320–340 |
4 |
|
340–360 |
6 |
|
360–380 |
8 |
|
380–400 |
12 |
|
400–420 |
15 |
|
420–440 |
5 |
|
440–460 |
5 |
|
460–480 |
3 |
Calculate a suitable measure of dispersion and justify your choice.
|
Bonus (in Rs.) |
No. of Workers |
c.f. |
|
300–320 |
2 |
2 |
|
320–340 |
4 |
6 |
|
340–360 |
6 |
12 |
|
360–380 |
8 |
20 |
|
380–400 |
12 |
32 |
|
400–420 |
15 |
47 |
|
420–440 |
5 |
52 |
|
440–460 |
5 |
57 |
|
460–480 |
3 |
60 |
Find out the quartile range, quartile deviation and coefficient of quartile deviation of the following data:
3, 7, 9, 13, 17, 17, 19, 20, 21, 24, 26
|
Sl. No. |
X |
|
1 |
3 |
|
2 |
7 |
|
3 |
9 |
|
4 |
13 |
|
5 |
17 |
|
6 |
17 |
|
7 |
19 |
|
8 |
20 |
|
9 |
21 |
|
10 |
24 |
|
11 |
26 |
|
ΣX= 11 |
Calculate mean deviation and coefficient of mean deviation of the following data:
Marks:45, 47, 47, 49, 50, 53, 58, 59, 60
|
Marks (X) |
|
|
45 |
7 |
|
47 |
5 |
|
47 |
5 |
|
49 |
3 |
|
50 |
2 |
|
53 |
1 |
|
58 |
6 |
|
59 |
7 |
|
60 |
8 |
|
ΣX = 468 |
Σd = 44 |
Ans :
In a town, 25% of persons earned more than Rs. 60,000 whereas 75% earned more than Rs. 20,000. Calculate the absolute value and relative value of dispersion.
(i) Absolute value of dispersion i.e.
(ii) Relative value of dispersion i.e.,
Draw a Lorenz Curve of the data given below:
|
Income (Rs.) |
No. of Persons |
|
100 |
80 |
|
200 |
70 |
|
400 |
50 |
|
500 |
30 |
|
800 |
20 |
|
Income |
Cumulative |
Cumulative |
No. of |
Cumulative |
Cumulative |
|
Income |
in pecentage |
Persons |
Persons |
percentage |
|
|
100 |
100 |
5 |
80 |
80 |
32 |
|
200 |
300 |
15 |
70 |
150 |
60 |
|
400 |
700 |
35 |
50 |
200 |
80 |
|
500 |
1500 |
60 |
30 |
230 |
92 |
|
800 |
2000 |
100 |
20 |
250 |
100 |
Draw a Lorenz Curve with the help of following data :
|
Wages |
No. of Workers |
|
50 – 70 |
20 |
|
70 – 90 |
15 |
|
90 – 110 |
20 |
|
110 – 130 |
25 |
|
130 – 150 |
20 |
|
Wages |
Wages (M.V.) |
Cumulative Sum |
Cumulative in percentage |
No. of workers |
c.f. |
c.f. in percentage |
|
50 – 70 |
60 |
60 |
12 |
20 |
20 |
20 |
|
70 – 90 |
80 |
140 |
28 |
15 |
35 |
35 |
|
90 – 110 |
100 |
240 |
48 |
20 |
55 |
55 |
|
110 – 130 |
120 |
360 |
72 |
25 |
80 |
80 |
|
130 – 150 |
140 |
500 |
100 |
20 |
100 |
100 |
Given below are the monthly incomes of employees of a company. Draw a Lorenz Curve with it. Also write down the steps required for drawing a Lorenz Curve.
|
Incomes |
Number of Employees |
|
0–5,000 |
5 |
|
5,000–10,000 |
10 |
|
10,000–20,000 |
18 |
|
20,000–40,000 |
10 |
|
40,000–50,000 |
7 |
|
Income limits |
Midpoints |
Cumulative mid-points |
Cumulative midpoints as percentages |
No. of employees frequencies |
Cumulative frequencies |
Cumulative frequencies as pecentages |
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
|
0-5000 |
2500 |
2500 |
2.5 |
5 |
5 |
10 |
|
5000-10000 |
7500 |
10000 |
10.0 |
10 |
15 |
30 |
|
10000-20000 |
15000 |
25000 |
25.0 |
18 |
33 |
66 |
|
20000-40000 |
30000 |
55000 |
55.0 |
10 |
43 |
86 |
|
40000-50000 |
45000 |
100000 |
100.0 |
7 |
50 |
100 |
Steps required for the construction of a Lorenz Curve : Following steps are required for the construction of a Lorenz Curve :
1. Calculate class mid-points and find cumulative totals as in Col. 3 in the questions, given above.
2. Calculate cumulative frequencies as in Col. 6.
3. Express the grand totals of Col. 3 and 6 as 100, and convert the cumulative totals in these columns into percentages, as in Col.4 and 7.
4. Now, on the graph paper, take the cumulative percentages of the variable (incomes) on Y axis and cumulative, percentages of frequencies (number of employees) on X-axis, as in figure. Thus each axis will have values from ‘0’ to ‘100’.
5. Draw a line joining Co-ordinate (0,0) with (100,100). This is called the line of equal distribution shown as line ‘OC’ in figure.
6. Plot the cumulative percentages of the variable with corresponding cumulative percentages of frequency. Join these points to get the curve OAC.
The coefficient of variation of two series are 58% and 69% and their standard deviation are 21.2 and 15.6. What are their mean?
(i) Coefficient of Variation of first series 
(ii) Coefficient of Variation of second series
The coefficient of variation of X series is 14.6% and that of Y series is 36.9% and their means are 101.2 and 101.25 respectively. Find their Standard Deviation.
(i) Coefficient of variation of X series 
(ii) Coefficient of variation of Y series
In a particular distribution quartile deviation is 15 marks and the coefficient of quartile deviation is 0.6. Find the lower and upper quartiles.
Hence, upper quartile = 30 and lower quartile is 10 marks.
You are given the following heights of boys and girls:
|
Boys |
Girls |
|
|
Number |
72 |
38 |
|
Average height in inches |
68 |
61 |
|
Variance of distribution in inches |
9 |
4 |
1. Calculate coefficient of variance.
2. Calculate whose height is more variable.
(a) Calculation of Coefficient of variance of boys: 
(b) Calculation of Coefficient of Variance of girls:
Height of boys is more variable as their coefficient variance is more.
The number of employee, wages per employee and the variance of wages per employee for two factories are given below :
|
ltems |
Factory A |
Factory B |
|
No of Employees |
50 |
100 |
|
Average wages per employee (Rs.) |
120 |
85 |
|
Variance of wages per day (Rs.) |
9 |
16 |
(i) In which factory is there greater variation in the distribution of wage per employee?
(ii) Suppose in a factory B, the wages of an employee are wrongly noted as Rs. 120 instead of Rs. 100. What would be the corrected variance of factory B?
Calculation of Coefficient of variation in factory A: 
Calculation of Coefficient of variation in factory B:
There is greater variance in distribution of wages in factory B.
Correcting Mean and Variance in factory B:
With an example, prove that the sum of the square of the deviations from arithmetic mean is least i.e. less than the sum of the squares of the deviations of observations taken from any other value.
a) Calculation of sum of the squares of the deviation from A.M. from an imaginary data:
|
X |
|
|
|
(n) |
(n)2 |
|
|
1 |
–2 |
4 |
|
2 |
–1 |
1 |
|
3 |
0 |
0 |
|
4 |
+ 1 |
1 |
|
5 |
+ 2 |
4 |
|
|
Σn2 = 10 |
Here, sum of the square of the deviations from A.M. is 3
... (i)
(b) Calculation of sum of the square of the deviation taken from any other value i.e. 2 (except A.M.)
|
X |
|
|
|
(n) |
(n)2 |
|
|
1 |
–1 |
1 |
|
2 |
0 |
0 |
|
3 |
+ 1 |
1 |
|
4 |
+ 2 |
4 |
|
5 |
+ 3 |
9 |
|
Σ |
Σn2 = 15 |
Here, sum of the square of the deviation taken from any other value except A. M. is 15 ... (ii)
From (i) and (ii) we come to know that the sum of the square of the deviations from arithmetic mean is less.
Write down the features of quartile deviation.
Features of quartile deviation:
1. It is rigidly defined.
2. It is easy to calcualte and simple to understand.
3. It does not depend on all values of the variables.
4. The units of measurement of the quartile deviation are the same as these of variables.
Calculate mean deviation from median from the following data:
|
X |
10 |
20 |
30 |
40 |
50 |
|
Y |
2 |
8 |
15 |
10 |
4 |
|
X |
f |
cf |
D |
fD |
|
10 |
2 |
2 |
20 |
40 |
|
20 |
8 |
10 |
10 |
80 |
|
30 |
15 |
25 |
0 |
0 |
|
40 |
10 |
35 |
10 |
100 |
|
50 |
4 |
39 |
20 |
80 |
|
Σf =39 |
ΣfD = 300 |
Calculate standard deviation from the following data:
|
S.No. |
1 |
2 |
3 |
4 |
5 |
|
Monthly |
400 |
600 |
900 |
1400 |
1200 |
|
Income (Rs.) |
|
SI. No. |
Monthly Income (Rs.) |
d |
d1 (d ÷ 100) |
d2 |
|
1 |
400 |
–500 |
–5 |
25 |
|
2 |
600 |
–300 |
–3 |
9 |
|
3 |
900 |
0 |
0 |
0 |
|
4 |
1400 |
500 |
5 |
25 |
|
5 |
1200 |
300 |
3 |
9 |
|
N = 5 |
AM = 900 |
Σ d1 = 0 |
Σd2 = 68 |
Given below are the marks obtained by the students of a class. Calculate mean deviation and its coefficient using median data.
17, 35, 38, 16, 42, 27, 19, 11, 40, 25 Ans. To determine median of a series, its items are arranged in ascending order as below:
|
SI. No |
Marks |
D (Deviation from Median) |
|
1 |
11 |
15 |
|
2 |
16 |
10 |
|
3 |
17 |
9 |
|
4 |
19 |
7 |
|
5 |
25 |
1 |
|
6 |
27 |
1 |
|
7 |
35 |
9 |
|
8 |
38 |
12 |
|
9 |
40 |
14 |
|
10 |
42 |
16 |
|
Σ D = 94 |
Find out mean deviation and coefficient of mean deviation from arithmetic mean from the following data:
|
Class-Interval |
0–10 |
10–20 |
20–30 |
30–40 |
40–50 |
|
Frequency |
2 |
4 |
6 |
4 |
2 |
|
SI. No. |
f |
(M.V.) x |
fx |
d |
fd |
|
0–10 |
2 |
5 |
10 |
20 |
40 |
|
10–20 |
4 |
15 |
60 |
10 |
40 |
|
20–30 |
6 |
25 |
150 |
0 |
0 |
|
30–40 |
4 |
35 |
140 |
10 |
0 |
|
40–50 |
2 |
45 |
90 |
20 |
40 |
|
N = 18 |
Σ fx = 450 |
Σ fd = 160 |
Calculate S.D. from the following data:
X= 10, 20, 30, 40, 50
|
SI. No. |
X |
X2 |
|
1 |
10 |
100 |
|
2 |
20 |
400 |
|
3 |
30 |
900 |
|
4 |
40 |
1600 |
|
5 |
50 |
2500 |
|
N = 5 |
Σ X= 150 |
Σ f x2 = 5500 |
Calculate standard deviation from the given values directly Le. without taking deviations. Values : 5, 10, 25, 30, 50.
Calculation of standard deviations directly i.e. without taking deviations:
|
X |
X2 |
|
5 |
25 |
|
10 |
100 |
|
25 |
625 |
|
30 |
900 |
|
50 |
2500 |
|
ΣX = 120 |
ΣX2 = 4150 |
Calculate range and co-efficient of range from the following data:
4, 7, 8, 46, 53, 77, 8, 1, 5, 13.
(i) Range= H-L = 77-l = 76
(ii) Co-efficient of Range
Find out quartile deviation and coefficient of quartile deviation of the following series.
Wages of 9 workers in Rs. are 170, 82, 110, 100, 150, 150, 200, 116, 250.
Rearranging the wages, we get
82, 100, 110, 116, 150, 150, 170, 200, 250
How is dispersion of the series different from average of the series?
Average of the series refers to central tendency of series whereas dispersion measures the extent to which different items tend to disperse away from the central tendency.
Name the methods of absolute measures of dispersion.
(i) Range (ii) Quartile deviation, (iii) Mean deviation, (iv) Standard deviation, (v) Lorenz curve.
What is the principal drawback of mean deviation as a measure of dispersion?
The principal drawback of mean deviation dispersion is that all deviations from the avarage value of the series are taken as positive, even when some of these are actually negative.
Give one point of difference between mean deviation and standard deviation.
In the calculation of mean deviation, deviation may be taken from mean, median or mode, but in the calculation of standard deviations are taken only from the mean value of the series.
What is difference between coefficient of variation and variance?
Co-efficient of variation is estimated
as 
whereas variance is the square of standard deviation 
How do range and quartile deviation measure the dispersion?
Range and quartile deviation measure the dispersion by calculating the spread within which the values lies.
What do mean deviation and standard deviation calculate?
Mean deviation and standard deviation calculate the extent to which the values differ from the average.
Name the measures which are based upon the spread of values.
(i) Range, (ii) Quartile deviation.
(i) Range, (ii) Quartile deviation.
(i) Range is unduly affected by extreme values.
(ii) It cannot be calculated for open-end distribution.
Name any two measures of dispersion from average.
(i) Mean deviation and (ii) Standard deviation.
When is the mean deviation the least and when is higher?
Mean deviation is the least when calculated from the median and will be higher if calculated from the mean.
Can standard deviation in case of individual series be calculated from the values directly i.e. without taking deviations while using the direct method? Write down the formula.
Yes. In this case following formula is used:
Write down any two merits and two demerits of mean deviation.
Two merits of mean deviation : 1. Mean deviation is less affected by extreme values than the range.
2. It can be calculated from any average (mean, median, mode)
Two demerits of mean deviation : 1. It is not capable of any further algebraic treatment 2. Calculation of mean deviation suffers from inaccuracy because the ‘+’ or ‘–’ signs are ignored.
Semi-inter quartile range is also known as:
Mean deviation
Standard deviation
Quartile deviation
Quartile range
C.
Quartile deviation
M.D. represents:
Mean deviation
Median deviation
Marginal deviation
None of above
B.
Median deviation
cf is used for:
Common factor
Cumulative frequency
Common value
None of above
B.
Cumulative frequency
Value of median is:
Second quartile
First quartile
Third quartile
Fourth quartile
A.
Second quartile
Median can be calculated from:
Individual series
Discrete series
Continuous series
All the above
D.
All the above
Which is the correct statement?
In the calculation of standard deviation, deviations may be taken from mean, median or mode.
In the calculation of standard deviations, signs of deviations (+) or (–) are ignored.
In the calculation of standard deviation, signs are not ignored.
In the calculation of mean deviation, deviations are taken only from the mean value of the series.
C.
In the calculation of standard deviation, signs are not ignored.
represents:
Lower quartile
First quartile
Both (a) and (b)
Neither (a) nor (b)
A.
Lower quartile
Decile is the division of the series into:
Two parts
Fifty parts
Ten parts
Hundred parts
B.
Fifty parts
Range is the:
Ratio of the largest to the smallest observations
Average of the largest to smallest observations
Difference between the smallest and the largest observations
Difference between the largest and the smallest observations
D.
Difference between the largest and the smallest observations
Write the false statement:
Range is the easiest and simplest measure because it is the difference between two extreme items.
Quartile deviation is superior to the range as it is not affected too much by the value of extreme items.
Range and quartile deviation are based on all the items of series
Range and quartile deviation are useful for general study of variability
C.
Range and quartile deviation are based on all the items of series
Which is not the relative measure of dispersion?
Co-efficient of range
Lorenz curve
Co-efficient of mean deviation
Co-efficient of variation
C.
Co-efficient of mean deviation
Define dispersion.
Dispersion measures the extent to which different items tend to disperse away from the central tendency.
How many methods are there to calculate dipersion?
Following are the methods of absolute and relative measures of dispersion :
(i) Absolute measure : Range, quartile deviation, ‘mean deviation, standard deviation, Lorenz curve.
(ii) Relative measure : Coefficient of range, coefficient of quartile deviation, coefficient of mean deviation, standard deviation, coefficient of variation.
Define range.
Range is the difference between the highest value and lowest value in a series.
Define quartile deviation.
Quartile deviation is half of Inter Quartile Range.
Quartile deviation =
How is coefficient of quartile deviation calculated?
Coefficient of quartile deviation is calculated by using the following formula:
Coefficient of QD
Define mean deviation.
Mean deviation is the arithmetic average of the deviations of all the values taken from some average value (mean, median, mode) of the series, ignoring sign (+ or –) of the deviations.
Define standard deviation.
Standard deviation is the square root of the arithmetic mean of the squares of deviations of the items from their mean value.
What is Lorenz Curve?
Lorenz Curve is a measure of deviation of actual distribution from the line of equal distribution.
What do you mean by coefficient of variation?
Coefficient of variation is a percentage expression of standard deviation. It is 100 times the coefficient of dispersion based on standard deviation of a statistical series.
Give formula of mean deviation through mean for individual series.
Mean Deviation through mean for individual series. 
What is standard deviation?
Standard deviation is the positive square root of the mean of squarred deviations from mean. S.D. is always calculated on the basis of mean only.
What is variance?
Variance is the square of standard deviation. In equation
Variance = (SD)2
Name the four methods available for the calculation of standard deviation of individual series.
(i) Actual mean method (ii) Assumed mean method (iii) Direct method and step deviation method.
What is dispersion?
The degree to which numerical data tend to spread about an average value is called the variation of dispersion. It is an average of second order.
What is measure of dispersion?
The measure of the deviation of the size of items from an average is called a measure of dispersion.
Name the important measures of dispersion.
Range, quartile deviation, mean deviation and standard deviation are the important measures of dispersion.
Define the range.
The range is defined as the difference between the largest and the smallest value of the variable in the given set of values.
R = L —S.
What is mean deviation or mean absolute deviation?
The arithmatic mean of the absolute deviation is called the mean deviation or mean
absolute deviation. Thus 
is the mean deviation of X about the arithmetic mean.
What is standard deviation?
The positive square root of the variance is called the standard deviation of the given value. In equation
Standard Deviation
Standard deviation is always positive. It is absolute measure.
What is the Variance?
Variance is the square of standard deviation. In equation, Variance = (σ)2
What is the difference between variance and standard deviation?
The variance is the average squared deviation from mean and standard deviation is the square root of variance.
Write down the unique feature of mean deviation.
Mean deviation is the least when taken about median.
Write down the unique feature of the variance.
The variance is unaffected by the choice of assumed mean.
What is coefficient of variation?
Coefficient of variation is the percentage variation in the mean, the standard being treated as the total variation in the mean.
What is a Lorenz Curve?
Lorenz Curve is a curve which measures the distribution of wealth and income. Now it is also used for the study of the distribution of profits, wages etc.
How do Range and Quartile deviation measure the dispersion?
Range and quartile deviation measure the dispersion by calculating the spread within which the values lie.
What do mean deviation and standard deviation calculate?
Mean deviation and standard deviation calculate the extent to which the values differ from the average.
Which aspect of distribution is indicated and which is not indicated by the averages?
Averages try to tell only one aspect of a distribution i.e. a representative size of the values. It does not tell us the spread of yalues.
Name the measures based on the spread of values.
Range and quartile deviation based on the spread of values.
Give two limitations of range.
1. Range is unduly affected by extreme values.
2. It is not based on all the values.
Not with standing some limitations. Why is range understand and used frequently?
Range is understand and used frequently because of its simplicity.
What is other name of quartile deviation?
The other name of quartile deviation is semi -inter quartile range.
Why is quartile deviation called semi inter - quartile range?
Quartile deviation is called semi-inter quartile range because it is half of the inter -quartile range.
Calculate the range of the following observations:
20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70
Range = L – S = 70 – 20= 50
Calculate quartile deviation from the information given below:
(i) Q3 = 59 (ii) Q1 = 29
Quartile deviation =
How is Q1 calculated in individual and discrete series?
In individual and discrete series. Q1 is calculated by adopting the following formula:
In continuous series, how is Q1 calculated?
In continuous series Q1 is calculated by applying formula.
Name the measures of dispersion from average.
Mean deviation and standard deviation are the measures of dispersion from average.
How is standard deviation independent of origin?
Standard deviation is independent of origin as it is not affected by the value of constant from which deviations are calculated. The value of the constant does not figure in the standard deviation formula.
What are open ended distribution?
Open ended distributions are those distribution in which either the lower limit of the lowest class or the upper limit of the highest class or both are not specified.
Which measure of dispersion is the best and how?
Standard deviation is the best measures of dispersion, because it posseses most of the characterstics of an ideal measure of dispersion.
Which type of measure is required for comparing the variability of two or more distribution given in different units of measurement?
Relative measure is required for comparing the variability of two or more distributions given in different units.
Calculate coefficient range of following distribution :
|
Maths |
No. of Students |
|
0 – 10 |
4 |
|
10 – 20 |
8 |
|
20 – 30 |
12 |
|
30 – 40 |
13 |
Coefficient of Range =
In a town 25% of person earned more than Rs. 45000 whereas 75% earned more than Rs. 18,000. Calculate absolute value of dispersion.
Absolute value of dispresion (quartile deviation) = 
Why is it better calculate of M.D. from median than that from mean?
It is better to calculate M.D. from median than that from mean because the sum of the deviations taken from median ignoring ± signs is less than the sum of deviations taken from mean.
Write down any one demerit of mean deviation.
Mean deviation cannot be computed with open end class.
How is mean deviation not well - defined measure?
Mean deviation is not well - defined measure because it is calculated from different averages ( Mean, median and mode) and mean deviation calculated from various averages will not be the same.
What is the difference between Variance and standard deviation?
The variance is the average squared deviation from mean and standard deviation is the square - root of variance.
What is absolute measure?
When dispersion of the series is expressed in terms of the original unit of the series, it is called absolute series.
What is the other name of relative measure of dispersion?
The other name of dispersion is coefficient of dispersion.
Write down the relative measures of standard deviation.
Relative measures of standard deviation are (i) Coefficient of standard deviation and (ii) Coefficient of variation.
Write down any one difference between mean deviation and standard deviation.
In the calculation of mean deviation, signs of deviations (+) or (–) are ignored, but in the calculation of standard deviation, signs are not ignored.
What is coefficient of variation?
Coefficient of variation is the percentage variation is the mean, the standard deviation being considered as the total variation in the mean.
How is coefficient of variation calculated?
Coefficient of variation is calculated by dividing the product of standard deviation or related and hundred by mean of the series.
Write down the formula of calculating coefficient if variation.
Coefficient of variation or C.V.
What does higher value of coefficient variation suggest?
Higher value of coefficient variation suggests greater degree of variability and less degree of stability.
What does lower value of coefficient variation suggest?
Lower value of coefficient variation suggest low degree of variability and higher degree of stability, unfromly, homogeneity and consistency.
If in the previous question, each worker is given a hike of 10% in wages, how are the mean and standard deviation values affected?
With the hike of 10% in wages, the mean will be Rs. 220 / (200 + 20)
There will be affected on standard deviation
Illustrate the meaning of the term dispersion with examples.
Dispersion is a measure of the variation of the items. According to Prof. C.R. Conn, Dispersion is a measure of the extent to which the individual item vary. The measures of dispersion are required to measures the amount of variation of values about the central values.
Example : Suppose the monthly incomes in rupees of five house holds are as Rs. 4500, 6000, 5500, 3750 and 4700.
The arithmatic mean of income is Rs. 4700. The amount of variation in income is shown by deviations from the central values. In this case the deviation from the arithmean are Rs. 390, 1110, 610, 1140 and 190.
What are the properties of a good measure of dispersion?
Properties of a good measure of dispersion:
1. It should be based on all the observations.
2. It should be readily comprehensible.
3. It should be fairly and easily understood.
4. It should be amendable to further algebric treatment.
5. It should be affected as little as possible by fluctuations in sampling.
Some measures of dispersion depend upon the spread of values whereas some calculate the variation of values from a central value. Do you agree?
Yes we agree with the statement Range and quartile deviation measure the dispersion by calculating the spread within which the values lie i.e. they depend on the spread of values. On the other hand, mean deviation and standard deviation calculate the variation of value from a central value.
In the previous question, calculate the relative measures of variation and indicate the value, which in your opinion is more reliable.
Co-efficient Of range is the relative measure of range. Hence we will calculate coefficient of range.
Co-efficient of Range of Wheat
Co-efficient of Range of Rice
In the same way, we will calculate co-efficient of quartile deviation and co-efficient of variation of both the crops.
Relative measure of variation is more reliable.
Average daily wage of 50 workers of a factory was Rs. 200 with a standard deviation of Rs. 40. Each worker is given a raise of Rs. 20. What is the new average daily wage and standard deviation? Have the wages become more or less uniform?
Increase in each worker wages = Rs. 20
Total increase in wages = 50 × 20 = Rs. 1000
Total of wages before increase worker in wages = 50 × 200 = Rs. 10,000
Total wages after increase in wages
Hence, mean wages will be affected but standard deviation will not be affected as the standard deviation is independent of origin. Have the wages become or less uniform? In order to calculate uniformity wages, we will have to calculate co-efficient of variation.
Afterwards
Now more uniformity in wages has taken place.
Give the absolute and relative measures of dispersion.
|
Absolute Measure |
Relative Measure |
|
1. Range |
1. Coefficient of range. |
|
2. Quartile deviation. |
2. Coefficient of Quartile deviation. |
|
3. Mean deviation |
3. Coefficient of Mean deviation. |
|
4. Standard deviation |
4. Coefficient of Standard deviation |
|
5. Lorenz curve. |
Write down the steps involved in the calculation of mean deviation in case of discrete series.
Steps involved in the calculation of mean deviation:Following steps are involved in the calculation of mean Deviation:
1. Find out the mean/median/mode of a series.
2. Find out the deviation of different items from mean/median/ mode.
3. Add the deviations ignoring positive and negative signs. Treat all deviations as positive.
4. Calculate mean deviation by dividing the sum total of the deviation by the number of items.
Write down the steps involved in the calculation of mean deviation for the discrete series.
Steps : 1. Find out central tendency of the series (mean or median) from which deviations are to be taken.
2. Take deviation of different items in the series from central tendency ignoring signs (+,–). Express it as | dx | or (| dm |).
3. Multiply each deviation value by frequency facing it.
4. Add the multiplies and express it as Σ?(d).
5. Divide Σ?(d) by sum total of frequency. The resultant value will be mean deviation.
Write down the features of mean deviation.
Features of mean deviation:
Tips: -
1. Mean deviation is rigidly defined.
2. It depends on all the values of the variable.
3. It is based on absolute deviations from central values.
4. It is easy to understand.
5. It involves harder calculation than the range and quartile deviation.
6. It is amendable to algebraic treatment.
7. The units of measurement of the mean deviation are the same as those of the variable.
Differentiate between Mean Deviation and Standard Deviation.
These are differences between Mean Deviation and Standard Deviation.
|
Mean Deviation |
Standard Deviation |
|
1. In calculating mean deviation. algebraic signs are ignored. |
1. In calculating standard deviation, algebraic signs are taken into account. |
|
2. Mean or median is used in calculating the mean deviation. |
2. Only mean is used in calculating the standard deviation. |
What are the uses of coefficient of variation?
Coefficient of variation is used to compare the variability, homogeneity, stability and uniformity of two different statistical series. Higher value of coefficient variation suggests greater degree of variation and lesser degree of stability. On the other hand, a lower value of coefficient variation suggests lower degree of variability and higher degree of stability, uniformity, homogeneity and consistency.
Explain merits and demerits of quartile Deviation.
Merits : 1. It is easy understand and to calculate.
2. It is unaffected by the extreme values.
3. It is quite satisfactory when only the middle half of the group is dealt with.
Demerits : 1. It ignores 50 per cent of the extreme items.
2. It is not capable of algebraic treatment.
3. This is not useful when extreme items are to be given special height.
Write down the merits of mean deviation.
Merits of Mean Deviation : 1. It is easy to understand mean Deviation.
2. Mean Deviation is less affected by extreme value than the Range.
3. Mean deviation is based on all the items of the series. It is therefore, more representative than the Range or Quartile Deviation.
4. It is very simple and easy measure of dispersion.
Demerits of Mean Deviation : 1. Mean deviation is not capable of algebraic treatment, because it ignores plus and minus signs.
2. It is not a well-defined measure since mean deviation from different averages (mean, median and mode) will not be the sameThe height of 11 men were 61, 64, 68, 69, 67, 68, 66, 70, 65, 67 and 72 inches. Calculate the range of the shortest man is omitted, what is the percentage change in the range?
1. Range = L = S
= 72 - 61 = 11 inches
2. New Range (Shortest man is omitted)
= L - S
= 72 - 64 = 8
Change in range = 11-8 = 3 inches
percentage change in range = 3/11 ×100 = 27.2%
What will be the effect of change of origin and change of scale on the standard deviation, mean and variance?
Change of origin and change of scale : Following are the effects of change of origin and change of scale on the mean, standard deviation and variance.
1. Any constant added or substracted (change of origin) than the standard deviation of original data and of change data after addition or substraction will not change but the mean of new data will change.
2. Any constant multiplied or divided (Change of scale) then mean, standard deviation and variation will change of the new changed data.
The following table gives you the height of 100 persons. Calculate dispersion by range method.
|
Height in Centimetres |
No. of Persons |
|
Below 162 |
2 |
|
Below 163 |
8 |
|
Below 164 |
19 |
|
Below 165 |
32 |
|
Below 166 |
45 |
|
Below 167 |
58 |
|
Below 168 |
85 |
|
Below 169 |
93 |
|
Below 170 |
100 |
Calculation of dispersion by Range method:
|
Height in Centimetres |
No. of Persons |
|
161–162 |
2 |
|
162–163 |
6 |
|
163–164 |
11 |
|
164–165 |
13 |
|
165–166 |
13 |
|
166–167 |
13 |
|
167–168 |
27 |
|
168–169 |
8 |
|
169–170 |
7 |
|
Total |
100 |
In a town 25% of the persons earned more than Rs. 45000, whereas 75% earned more than Rs. 18000. Calculate the absolute and relative values of dispersion.
1. Absolute value of dispersion i.e. 
How is dispersion of the series different from average of the series?
Average of the series refers to the central tendency of the series. It represents behaviour of all the items in the series. But different items tend to different from each other and from the averages. Dispersion measures the extent to which different items tend to disperse away from the central tendency.
Why should we measure dispersion about some particular value?
We should measure dispersion about some particular value because in that case (i) We can assess how precise is the central tendency as the representative value of all the observations in the series. Greater value of dispersion implies lesser representativeness of the central tendency and vice versa.
(ii) We can precisely asses how scattered are the actual observation from their representative value.
Why is standard deviation also known as the root mean square deviation?
Standard deviation is also known as the root mean square deviation because it is the square root of the means of the square deviation from the arithmetic mean. In the calculation of standard deviation, first the arithmetic average is calculated, and the variations of various items from arithmetic averages are squared. The squared deviations are totalled and the sum is divided by the number of items. The square root of the resulting figure is the standard deviation of the series. The S.D. is denoted by the Greek letter. σ (Sigma) Symbalically
Comparison of alternative measures of the dispersion has been discussed below:
1. Rigidly defined : All the four measures-the range, quartile deviation, mean deviation and standard deviation are rigidly defined. There is no vagueness in their definition.
2. Ease of calculation : The range is the easiest one to calculate. Quartile deviation requires calculation of the upper and lower quartiles but that is also easy enough. However, the mean deviation and standard deviation require a little more systematic calcualtion. They, too are easy.
3. Simple interpretation : All measures of dispersion are easy to interpret. While the range and quartile deviation measure dispersion in a general way the mean deviation and standard deviation measure dispersion in terms of deviations from a central value. Thus the mean deviation and standard deviation give a better idea about the dispersion of values within the range.
4. Based on all values : The range and quartile deviation do not depend on all values, whereas, the mean deviation and standard deviation use all values of the variable. The range is affected the most by extreme values.
5. Amendable to algebraic treatment : The standard deviation is perhaps the easiest for analytical work. Other measures can be also dealt with analytically but derivation are harder.
What are the four alternative measures of absolute dispersion ? Discuss their properties.
The four alternative measure of absolute dispersion are:
(i) Range, (ii) Quartile Deviation, (iii) Mean deviation, (iv) Standard Deviation.
1. Features of the range :
(a) It is rigidly defined.
(b) It is easy to calculate and simple to interpret.
(c) It does not depend on all values of the variables.
(d) It is unduly affected by extreme values.
(e) The range depends on the units of measurement of the variable.
2. Features of Quartile deviation :
(a) It is rigidly defined.
(b) It is easy to calculate and simple to interpret.
(c) It does not depend on all values of the variable.
(d) The units of measurement of the quartile deviation are the same as those of the variable.
3. Features of mean deviation :
(a) It is rigidly defined.
(b) It depends on all values of the variable.
(c) It is based on absolute deviations from a central values.
(d) It is easy to understand.
(e) It involves harder calculations than the range and quartile deviation.
(f) It is amendable to algebraic treatment.
(g) The units of measurement of the mean deviation are the same as those of the variable.
4. Features of Standard deviation :
(a) It is the best measure of the variation because it is based every item of the series and further algebraic treatment is possible.
(b) It is not very much affected by fluctuation of sampling.
(c) It is the only measure for calculating combined standard deviation of two or more graphs.
(d) It is a definite measure of dispersion.
“ The coefficient of variation is a relative measure of disperison’. We may calculate coefficient of variation using any of the measure of dispersion such as range, quartile deviation, mean deviation and standard deviation.
Illustrate the use of coefficient of variation in these cases.
There are two types of dispersion absolute measure and relative measures of dispersion. Absolute measures of dispersion are measured in the same units as those of variables considered. This feature of measures of dispersion may create difficulty if we want to compare dispersion in two sets of values which have (i) different central values and (ii) different units of measurement.
In order to overcome this difficulty it is desirable to eliminate the units. This can be done if we use a relative measure of dispersion which is a pure number and do not depend on units of measurement. The relative measure of dispersion is called the cofficient of variation. It may be expressed as ratio or express it in percentage.
The most commonly used coefficient of variation is ratio of standard deviation we may also express in percentage as
Where σ is the standard deviation and m is arithmetic mean.
We may also compute the coefficient of variation as
if we are using the range as measure of. dispersion.
if we are using quartile deviation as measure of dispersion.
Similarly using mean deviation,
Give the formulae of range, Quartile deviation, mean deviation, standard deviation (Absolute and relative measures both.)
1. Range
Range = L – S
L = Largest item
S = Smallest item
2. Quartile deviation
Q1 = Lower Quartile.
Q3 = Upper Quartile.
3. Mean Deviation
Individual Observation :
4. Standard Deviation
Individual observators
Actual Mean Method
An analysis of the weekly wages paid to workers in two firms A and B give the following result:
|
Firm A |
Firm B |
|
|
No. of Workers |
586 |
648 |
|
Average weekly wages |
Rs. 52.5 |
Rs. 47.5 |
|
Variance of the distribution of wages |
100 |
121.0 |
Which firm “A” or “B” has greater variability in individual wage.
Since coefficient of variation is higher in case of firm B, it shows greater variability in individual wages.
Prove by an example that the variance is unaffected by the choice of the assumed mean.
We take the following example for proving that the variance is uneffected by the choice of the assumed mean.
Example : Calculate variance of 25, 50, 45, 30, 70, 42, 36, 48, 34 and 60 by actual mean assumed mean method.
(a) Calculation of Variance by Actual Mean Method
|
Values of X |
|
x2 |
|
25 |
–19 |
361 |
|
50 |
+6 |
36 |
|
45 |
+1 |
1 |
|
30 |
–14 |
196 |
|
70 |
–26 |
676 |
|
42 |
–2 |
4 |
|
36 |
–8 |
64 |
|
48 |
+4 |
16 |
|
34 |
–10 |
100 |
|
60 |
+16 |
256 |
|
ΣX = 440 |
ΣX2 = 1710 |
(b) Calculation of Variance by Assumed Mean Method
|
Values X |
(X–45) d |
d2 |
|
25 |
–20 |
400 |
|
50 |
+5 |
25 |
|
45 |
0 |
0 |
|
30 |
–15 |
225 |
|
70 |
+25 |
625 |
|
42 |
–3 |
9 |
|
36 |
–9 |
81 |
|
48 |
+3 |
9 |
|
34 |
–11 |
121 |
|
60 |
+15 |
225 |
|
N=10 |
Σd = –10 |
Σd2 =1720 |
The distribution of the cost of production (in rupees) of a quintal of wheat in 50 farms is as follows:
|
Cost (in Rs.) |
Number of Farms |
|
40–50 |
3 |
|
50–60 |
6 |
|
60–70 |
12 |
|
70–80 |
18 |
|
80–90 |
9 |
|
90–100 |
2 |
|
Total |
50 |
(a) Calculate the variance.
(i) by direct method.
(ii) by step deviation method and compare your results with the mean deviation about the arithmatic mean.
(b) Calculate the coefficient of variation by using
(i) the standard deviation of costs and
(ii) the mean deviation of cost about the arithmatic mean and compare the two. What is your conclusion about variation of cost.
Calcualtion of Variance by Direct Method.
|
Class Interval |
f |
Mid Point |
fx |
|
fd |
d2 |
fd2 |
|
40-50 |
3 |
45 |
135 |
-26 |
78 |
676 |
2028 |
|
50-60 |
6 |
55 |
330 |
-16 |
96 |
256 |
1536 |
|
60-70 |
12 |
65 |
780 |
-6 |
72 |
36 |
432 |
|
70-80 |
18 |
75 |
1350 |
4 |
72 |
16 |
288 |
|
80-90 |
9 |
85 |
765 |
14 |
126 |
196 |
1764 |
|
90-100 |
2 |
95 |
190 |
24 |
48 |
576 |
1152 |
|
Σf = 50 |
Σf = 3550 |
Σfd2 = 7200 |
Calculation of Variance by Step Deviation
|
Class |
? |
Mid |
d |
||||
|
Interval |
Point (X) |
d’ |
d’2 |
?d' |
?d'2 |
||
|
40-50 |
3 |
45 |
–30 |
–3 |
9 |
–9 |
27 |
|
50-60 |
6 |
55 |
–20 |
–2 |
4 |
–12–33 |
24 |
|
60-70 |
12 |
65 |
–10 |
–1 |
1 |
–33 |
–12 |
|
70-80 |
18 |
75 |
0 |
0 |
0 |
0 |
0 |
|
80-90 |
9 |
85 |
10 |
+1 |
1 |
9+13 |
9 |
|
90-100 |
2 |
95 |
20 |
+2 |
4 |
4 |
8 |
|
Σ? = 80 |
Σd = 90 |
Σfd' = 20 |
Σfd'2 = 80 |
(b) Calculation of coefficient of variation :
1. Variance coefficient (From S.D.)
Briefly explain the various measures calculated from standard deviation.
Measures calculated from standard deviation:
Mainly following measures are calculated from standard deviation:
1. Coefficient of standard deviation : It is a relative measure of standard deviation. It is calculated to compare the variability in two or more than two series. It is calculated by dividing the standard deviation by arithmetic mean of data symbolically.
2. Coefficient of Variance : It is most propularly used to measure relative variation of two or more than two series. It shows the relationship between the S.D. and the arithmetic mean expressed in terms of percentage. It is used to compare uniformly, consistency and variability in two different series.
3. Variance : It is the square of standard deviation. It is closely related to standard deviation. It is the average squared deviation from mean where as standard deviation is the square is the square root of variance. Symbolically
A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are:
|
X |
Y |
|
25 |
50 |
|
85 |
70 |
|
40 |
65 |
|
80 |
45 |
|
120 |
80 |
(a) Calculate coefficient of standard deviation, variance and coefficient of variation.
(b) Which batsman should be selected if we wants.
(i) a higher run getter, or
(ii) a more reliable batsman in the team?
(a)
|
Scores |
Batsman x |
|
|
|
|
|
|
25 |
–45 |
2025 |
|
85 |
+15 |
225 |
|
40 |
–30 |
900 |
|
80 |
+10 |
100 |
|
120 |
+50 |
2500 |
|
ΣX=350 |
Σx2= 570 |
|
Batsman Y
|
Scores |
|
|
|
50 |
–12 |
144 |
|
70 |
8 |
64 |
|
65 |
3 |
9 |
|
45 |
–17 |
289 |
|
80 |
18 |
324 |
|
Σx=310 |
Σx2= 830 |
(i) Batsman X should be selected as a higher run getter as his average score (70 runs) is greater than that of Y (i.e. 62 runs)
(ii)Batsman Y is a more reliable batsman in the team because his coefficient of variance (20.77) is less than that of batsman X (c.v. 48.44)
To check the quality of two brands of lightbulbs, their life in burning hours was estimated as under for 100 bulbs of each brand.
|
Life (in hrs) |
No. of bulbs |
||
|
Brand A |
Brand B |
||
|
0–50 |
15 |
2 |
|
|
50–100 |
20 |
8 |
|
|
100–150 |
18 |
60 |
|
|
150–200 |
25 |
25 |
|
|
200–250 |
22 |
5 |
|
|
100 |
100 |
||
(i) Which brand gives higher life?
(ii) Which brand is more dependable?Brand A of light bulbs
|
Life |
No. of |
Mid-points |
d |
d1 |
fd' |
fd2 |
|
|
(in hrs) |
Bulbs (f) |
(m) |
(m – 125) |
|
|||
|
0–50 |
15 |
25 |
–100 |
–2 |
–30 |
60 |
|
|
50–100 |
20 |
75 |
–50 |
–1 |
–20 |
20 |
|
|
100–150 |
18 |
125 |
0 |
0 |
0 |
0 |
|
|
150–200 |
25 |
175 |
50 |
+1 |
25 |
25 |
|
|
200–250 |
22 |
250 |
100 |
2 |
44 |
88 |
|
|
N = 100 |
Σfd' = 19 |
Σfd2 =193 |
|||||
Brand B
|
Life |
No. of |
M.V. |
d |
|
fd' |
fd'2 |
|
|
(in hrs) |
Bulbs |
(m) |
d1 |
||||
|
0–50 |
2 |
25 |
–100 |
–2 |
–4 |
8 |
|
|
50–100 |
8 |
75 |
–50 |
–1 |
–8 |
8 |
|
|
100–150 |
60 |
125 |
0 |
0 |
0 |
0 |
|
|
150–200 |
25 |
175 |
+50 |
+1 |
+25 |
25 |
|
|
200–250 |
5 |
225 |
+100 |
+2 |
+10 |
20 |
|
|
N= 100 |
Σfd' =23 |
Σfd'2 = 61 |
|||||
(i) Since the average life of bulbs of Brand B (136.5) is greater than that of Brand A (134.5 hrs), therefore the bulbs of Brand B givens a higher life.
(ii) Since CV of bulbs of Brand B (27.34%) is less than that of Brand A (51.15%), therefore, the bulbs of Brand B are more dependable.
Calculate the standard deviation of the following values by following methods:
(i) Actual Mean Method, (ii) Assumed Mean Method, (iii) Direct Method, (iv) Step Deviation Method.
5, 10, 25, 30, 50.
(i) Calculation of Standard Deviation by Actual Mean Method :
|
X |
d |
d2 |
|
5 |
–19 |
361 |
|
10 |
–14 |
196 |
|
25 |
+1 |
1 |
|
30 |
+6 |
36 |
|
50 |
+26 |
676 |
|
ΣX = 120 |
0 |
Σd2 = 1270 |
(ii) Calculation of Standard Deviation by Assumed Mean Method :
|
X |
d |
d2 |
|
5 |
–20 |
400 |
|
10 |
–15 |
225 |
|
25 |
0 |
0 |
|
30 |
+5 |
25 |
|
50 |
+25 |
625 |
|
–5 |
1275 |
(iii) Calculation of Standard Deviation by Direct Method : Standard Deviation can also be calculated from the values directly, i.e., without taking deviations, as shown below :
|
X |
x2 |
|
5 |
25 |
|
10 |
100 |
|
25 |
625 |
|
30 |
900 |
|
50 |
2500 |
|
ΣX = 120 |
ΣX2 = 4150 |
(iv) Calculation of Standard Deviation by Step Deviation Method : The values are divisible by a common factor, they can be so divided and standard deviation can be calculated from the resultant values as follows :
Since all the five values are divisible by a common factor 5, we divide and get the following values :
|
x |
x2 |
d |
d2 |
|
5 |
1 |
–3.8 |
14.44 |
|
10 |
2 |
–2.8 |
7.84 |
|
25 |
5 |
+0.2 |
0.04 |
|
30 |
6 |
+1.2 |
1.44 |
|
50 |
10 |
+5.2 |
27.04 |
|
N = 5 |
0 |
50.80 |
Alternative Method : Alternatively, instead of dividing the values by a common factor, the deviations can be divided by a common factor. Standard Deviation can be calculated as shown below :
|
x |
d |
d' |
d2 |
|
5 |
–20 |
–4 |
16 |
|
10 |
–15 |
–3 |
9 |
|
25 |
0 |
0 |
0 |
|
30 |
+5 |
+1 |
1 |
|
50 |
+25 |
+5 |
25 |
|
N = 5 |
–1 |
51 |
Deviations have been calculated from an arbitrary value 25. Common factor of 5 has been used to divide deviations.
Calculate Mean Deviation from the following table using:
(i) Actual Mean Method
(ii) Assumed Mean Method
(iii) Step Deviation Method
|
Profits of Companies (Rs. in lakhs) |
Number of |
|
Class-intervals |
Companies frequencies |
|
10 – 20 |
5 |
|
20 – 30 |
8 |
|
30 – 50 |
16 |
|
50 – 70 |
8 |
|
70 – 80 |
3 |
|
40 |
(i) Calculation of S.D. with the help of Actual Mean Method :
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
|
CI |
f |
m |
fm |
d |
fd |
fd2 |
|
10–20 |
5 |
15 |
75 |
–25.5 |
–127.5 |
3251.25 |
|
20–30 |
8 |
25 |
200 |
–15.5 |
–124.0 |
1922.00 |
|
30–50 |
16 |
40 |
640 |
–0.5 |
8.0 |
4.00 |
|
50–70 |
8 |
60 |
480 |
+19.5 |
+156.0 |
3042.00 |
|
70–80 |
3 |
75 |
225 |
+34.5 |
+103.5 |
3570.75 |
|
Σf=40 |
Σfm=1620 |
Σfd=0 |
Σfd2= 11790.00 |
(ii) Calculation of Standard Deviation by Assumed Mean Method :
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
|
CI |
f |
m |
d |
fd |
fd2 |
|
10–20 |
5 |
15 |
–25 |
–125 |
3125 |
|
20–30 |
8 |
25 |
–15 |
–120 |
1800 |
|
30–50 |
16 |
40 |
0 |
0 |
0 |
|
50–70 |
8 |
60 |
+20 |
160 |
3200 |
|
70–80 |
3 |
75 |
+35 |
105 |
3675 |
|
Σf 40 |
Σfd=+20 |
Σfd2=11800 |
(iii) Calculation of Standard Deviation by Step Deviation Method :
|
(1) |
(2) |
(3) |
(4) |
(5) |
(6) |
(7) |
|
CI |
f |
m |
d |
d' |
fd' |
fd'2 |
|
10–20 |
5 |
15 |
–25 |
–5 |
–25 |
125 |
|
20–30 |
8 |
25 |
–15 |
–3 |
-24 |
72 |
|
30–50 |
16 |
40 |
0 |
0 |
0 |
0 |
|
50-70 |
8 |
60 |
+20 |
+4 |
+32 |
128 |
|
70–80 |
3 |
75 |
+35 |
+7 |
+21 |
147 |
|
40 |
+4 |
472 |
The Standard Deviation of height measured in inches will be larger than the Standard Deviation of the height measured in ft. for the same group of individuals. Comment on the validity or otherwise of the statement with appropriate illustration.
The statement is totally valid. The least is that Standard Deviation is an absolute measure. When the units of measurement are different the less the measurement will, the more will the Standard Deviation. With the increase in the measurement unit, the Standard Deviation will decrease. It is clear from the following illustrations.
Suppose we are given the heights of 5 persons in feet. such as 2, 4, 6, 8, 10. With the help of the data we will calculate the S.D.
|
Height in feet |
D |
D2 |
|
(X) |
|
|
|
2 |
–4 |
16 |
|
4 |
–2 |
4 |
|
6 |
0 |
0 |
|
8 |
2 |
4 |
|
10 |
4 |
16 |
|
ΣX = 30 |
ΣD2 = 40 |
Now we calculate S.D. taking the heights of 5 same persons in inches.
|
Height in inches |
D |
D2 |
|
(X) |
|
|
|
24 |
–48 |
2304 |
|
48 |
–24 |
576 |
|
72 |
0 |
0 |
|
96 |
+24 |
576 |
|
120 |
+48 |
|
|
ΣX = 360 |
ΣD2 = 5760 |
Thus, we see that S.D. has increased 12 times.
A study of certain examination results of 1000 students at the year 2000 gave average marks secured as 50% with a standard deviation of 3%. A similar study of 2001 revealed average marks secured and standard deviation as 55% and 5% respectively. Have the results improved?
Results have not improved in 2001 as the relative dispersion in 2001 is more than that of in 2000.
The yield of wheat and rice per acre for 10 districts of a state is as under:
|
District |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
|
Wheat |
12 |
10 |
15 |
19 |
21 |
16 |
18 |
9 |
25 |
10 |
|
Rice |
22 |
29 |
12 |
23 |
18 |
15 |
12 |
34 |
18 |
12 |
Calculate for each crop :
(i) Range, (ii) Q. D., (iii) Mean Deviation about Mean, (iv) Mean Deviation about Median, (v) Standard Deviation, (vi) Which crop has greater variation ?(vii) Compare the values of different measures for each cropTry Yourself.
Prove with an example that Q. D. is the average difference of the quartiles from Median.
In order to prove that Q. D. is the average difference of the quartiles from median. We calculate Q1, Q3, Q.D and median from the following data :
20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70.
Hence, it is proved Q.D. (here 11) is the average difference of the quartiles from the median.
Prove that mean deviation calculated about mean will be greater than that calculate about median.
In order to prove the statement given in the question we calculate mean deviation about mean and mean deviation from median and compare what is greater:
2, 4, 7, 8 and 9.
Mean deviation about Mean
|
X |
D |
|
2 |
4 |
|
4 |
2 |
|
7 |
1 |
|
8 |
2 |
|
9 |
3 |
|
ΣX = 30 |
ΣD = 12 |
|
X |
D (X–7) |
|
2 |
5 |
|
4 |
3 |
|
7 |
0 |
|
8 |
1 |
|
9 |
2 |
|
N = 5 |
ID = 11 |
Mean deviation about mean is 2.4. and mean deviation about median is 2.2. Hence, proved that mean deviation about mean is greater than mean deivation about median.
Prove that mean deviation is based on all values. A change in even one value will effect of.
In order to prove the statement given in the question, we calculate the mean deviation of the following data : 2, 4, 7, 8, 9.
|
X |
D |
|
2 |
4 |
|
4 |
2 |
|
7 |
1 |
|
8 |
2 |
|
9 |
3 |
|
ΣX = 30 |
ΣD = 12 |
Now change one value i.e. we take 14 in place of 9.
|
X |
D |
|
2 |
5 |
|
4 |
3 |
|
7 |
0 |
|
8 |
1 |
|
14 |
7 |
|
ΣX = 35 |
ΣD = 16 |
In this way we see that mean deviation changes with a change in even one value. Earlier, the mean deviation was 2.4. After the change in one value it is 3.2.
A measure of dispersion is a good .supplement to the central value in understanding a frequency distribution. Comment.
A measure of dispersion : A good supplement to the central value : A central value condenses the series into a single figure. The measure of central tendencies indicate the central tendency of a frequency distribution in the form of an average. These averages tell us something about the general level of the magnitude of the distribution, but they fail to show anything further about the distribution. The averages represent the series as a whole. One may now be keen to know how far the various values of the series tend to dispense from each other or from their averages. This brings us to yet another important brand of statistical methods, viz. measures of dispersion. Only when we study dispersion alongwith average of series that we can have a comprehensive information about the nature and composition of a statistical series.
In a country, the average income or wealth may be equal. Yet there may be great disparity in its distribution. As a result, thereof, a majority of people may be below poverty line. There is need to measure variation in dispersion and express it as a single figure. It can be further explained with an example. Below are given the family’s incomes of Ram, Rahim and Maria. Ram, Rahim and Maria have four, six and five members in their families respectively.
Family Income
|
St. No. |
Ram |
Rahim |
Maria |
|
1. |
12,000 |
7,000 |
— |
|
2. |
14,000 |
10,000 |
7,000 |
|
3. |
16,000 |
14,000 |
8,000 |
|
4. |
18,000 |
17,000 |
10,000 |
|
5. |
— |
20,000 |
50,000 |
|
6. |
— |
22,000 |
— |
|
Total |
60,000 |
90,000 |
75,000 |
From the table we come to know that each family have average income of Rs. 15,000
considerable differences in individual methods. It is quite obvious that averages try to tell only one aspect of a distribution i.e. representative size of the values. To understand it better, we need to know the spread of values also. The Ram’s family, differences in incomes are comparatively lower. In Rahim’s family, differences are higher and Maria’s family differences are the highest. Knowledge of only average is sufficient. A measure of dispersion improves the understanding of the distribution series.
How is dispersion of the series different from the average of the series? What will be the effect of change of origin and change of scale on S.D. mean and variance series?
(a) Difference between dispersion of series and average of series : Averages of series in known as the measures of central tendency. An average indicates respresentative value of the series around which other value of the series tend to converage. So the average represents the series as a whole. In the other hand dispersion is the measure of the variationes of the items. It helps us in knowing about the composition of a series or the dispersal of values on the either side of the central tendency.
(b) Effect of change of origin and change of scale in the S.D. mean and variance : Change of origin i.e. any constant added or subtracted will have no effect on standard deviation but it will change the mean.
On the other hand change of scale (any constant multiplied or divided) will change the mean, standard deviation and variance.
The heights of 11 men are 61, 64, 68, 69, 67, 68, 66, 70, 65, 67 and 72 inches. Calculate range of the man of the least height in removed. What will be the percentage change in the range?
1. Range =L – S
= 72 – 61 = 11 inches.
2. New range (after removing the man of least height) = 72 – 64 = 8 inches.
3. Change in range = 11 – 8 = 3 inches.
Percentage change in Range
Below is given the height of 100 men. Calculate dispersion by range method.
|
Height in Centimetres |
No. of Persons |
|
Less than 162 |
2 |
|
Less than 163 |
8 |
|
Less than 164 |
19 |
|
Less than 165 |
32 |
|
Less than 166 |
45 |
|
Less than 166 |
58 |
|
Less than 167 |
85 |
|
Less than 168 |
93 |
|
Less than 169 |
100 |
Calculation of Dispersion by Range Method:
|
Height in Centimetres |
No. of Persons |
|
161 – 162 |
22 |
|
162 – 163 |
6 |
|
163 – 164 |
11 |
|
164 – 165 |
13 |
|
165 – 166 |
13 |
|
166 – 167 |
13 |
|
167 – 168 |
27 |
|
168 – 169 |
8 |
|
169 – 170 |
7 |
|
Total |
100 |
Calculate the inter quartile range, quartile deviation and coefficient of quartile deviation of the following frequency distribution relating to bonus paid up to workers.
|
Bonus (in Rs.) |
No. of Workers |
|
300–320 |
2 |
|
320–340 |
4 |
|
340–360 |
6 |
|
360–380 |
8 |
|
380–400 |
12 |
|
400–420 |
15 |
|
420–440 |
5 |
|
440–460 |
5 |
|
460–480 |
3 |
Calculate a suitable measure of dispersion and justify your choice.
|
Bonus (in Rs.) |
No. of Workers |
c.f. |
|
300–320 |
2 |
2 |
|
320–340 |
4 |
6 |
|
340–360 |
6 |
12 |
|
360–380 |
8 |
20 |
|
380–400 |
12 |
32 |
|
400–420 |
15 |
47 |
|
420–440 |
5 |
52 |
|
440–460 |
5 |
57 |
|
460–480 |
3 |
60 |
Find out the quartile range, quartile deviation and coefficient of quartile deviation of the following data:
3, 7, 9, 13, 17, 17, 19, 20, 21, 24, 26
|
Sl. No. |
X |
|
1 |
3 |
|
2 |
7 |
|
3 |
9 |
|
4 |
13 |
|
5 |
17 |
|
6 |
17 |
|
7 |
19 |
|
8 |
20 |
|
9 |
21 |
|
10 |
24 |
|
11 |
26 |
|
ΣX= 11 |
Calculate mean deviation and coefficient of mean deviation of the following data:
Marks : 45, 47, 47, 49, 50, 53, 58, 59, 60
|
Marks (X) |
|
|
45 |
7 |
|
47 |
5 |
|
47 |
5 |
|
49 |
3 |
|
50 |
2 |
|
53 |
1 |
|
58 |
6 |
|
59 |
7 |
|
60 |
8 |
|
ΣX = 468 |
Σd = 44 |
In a town, 25% of persons earned more than Rs. 60,000 whereas 75% earned more than Rs. 20,000. Calculate the absolute value and relative value of dispersion.
(i) Absolute value of dispersion i.e.
(ii) Relative value of dispersion i.e.,
Draw a Lorenz Curve of the data given below:
|
Income (Rs.) |
No. of Persons |
|
100 |
80 |
|
200 |
70 |
|
400 |
50 |
|
500 |
30 |
|
800 |
20 |
|
Income |
Cumulative |
Cumulative |
No. of |
Cumulative |
Cumulative |
|
Income |
in pecentage |
Persons |
Persons |
percentage |
|
|
100 |
100 |
5 |
80 |
80 |
32 |
|
200 |
300 |
15 |
70 |
150 |
60 |
|
400 |
700 |
35 |
50 |
200 |
80 |
|
500 |
1500 |
60 |
30 |
230 |
92 |
|
800 |
2000 |
100 |
20 |
250 |
100 |
In a particular distribution quartile deviation is 15 marks and the coefficient of quartile deviation is 0.6. Find the lower and upper quartiles.
You are given the following heights of boys and girls:
|
Boys |
Girls |
|
|
Number |
72 |
38 |
|
Average height in inches |
68 |
61 |
|
Variance of distribution in inches |
9 |
4 |
1. Calculate coefficient of variance.
2. Calculate whose height is more variable.
(a) Calculation of Coefficient of variance of boys: 
(b) Calculation of Coefficient of Variance of girls :
Height of boys is more variable as their coefficient variance is more.
The number of employee, wages per employee and the variance of wages per employee for two factories are given below:
|
ltems |
Factory A |
Factory B |
|
No of Employees |
50 |
100 |
|
Average wages per employee (Rs.) |
120 |
85 |
|
Variance of wages per day (Rs.) |
9 |
16 |
(i) In which factory is there greater variation in the distribution of wage per employee?
(ii) Suppose in a factory B, the wages of an employee are wrongly noted as Rs. 120 instead of Rs. 100. What would be the corrected variance of factory B?
Calculation of Coefficient of variation in factory A: 
There is greater variance in distribution of wages in factory B.
Correcting Mean and Variance in factory B:
With an example, prove that the sum of the square of the deviations from arithmetic mean is least i.e. less than the sum of the squares of the deviations of observations taken from any other value.
(a) Calculation of sum of the squares of the deviation from A.M. from an imaginary data:
|
X |
|
|
|
(n) |
(n)2 |
|
|
1 |
–2 |
4 |
|
2 |
–1 |
1 |
|
3 |
0 |
0 |
|
4 |
+ 1 |
1 |
|
5 |
+ 2 |
4 |
|
|
Σn2 = 10 |
Here, sum of the square of the deviations from A.M. is 3
... (i)
(b) Calculation of sum of the square of the deviation taken from any other value i.e. 2 (except A.M.)
|
X |
|
|
|
(n) |
(n)2 |
|
|
1 |
–1 |
1 |
|
2 |
0 |
0 |
|
3 |
+ 1 |
1 |
|
4 |
+ 2 |
4 |
|
5 |
+ 3 |
9 |
|
Σ |
Σn2 = 15 |
Here, sum of the square of the deviation taken from any other value except A. M. is 15 ... (ii)
From (i) and (ii) we come to know that the sum of the square of the deviations from arithmetic mean is less.
Write down the features of quartile deviation.
Features of quartile deviation:
1. It is rigidly defined.
2. It is easy to calcualte and simple to understand.
3. It does not depend on all values of the variables.
4. The units of measurement of the quartile deviation are the same as these of variables.
Calculate the mean deviation about mean and standard deviation for the following distribution:
|
Classes |
Frequencies |
|
20–40 |
3 |
|
40–80 |
6 |
|
80–100 |
20 |
|
100–120 |
12 |
|
120–140 |
9 |
|
50 |
Calculation of mean deviation about mean :
|
Classes |
f |
mid value (m) |
fx |
|
fd |
|
20–40 |
3 |
30 |
90 |
64.8 |
194.4 |
|
40–80 |
6 |
60 |
360 |
34.8 |
208.8 |
|
80–100 |
20 |
90 |
1800 |
4.8 |
96.0 |
|
100–120 |
12 |
110 |
1320 |
15.2 |
182.4 |
|
120–140 |
9 |
130 |
1170 |
35.2 |
316.8 |
|
N = 50 |
Σfx = 4740 |
Σfd = 998.4 |
Calculation of standard deviation from mean:
|
Classes |
f |
m |
m–90 |
|
fd' |
fd'2 |
|
(d) |
(d') |
|||||
|
20–40 |
3 |
30 |
–60 |
–6 |
–18 |
108 |
|
40–80 |
6 |
60 |
–30 |
–3 |
–18 |
54 |
|
80–100 |
20 |
90 |
0 |
0 |
0 |
0 |
|
100–120 |
12 |
110 |
+20 |
+2 |
24 |
48 |
|
120–140 |
9 |
130 |
+40 |
+4 |
36 |
144 |
|
N = 50 |
Σfd' = 24. |
Σfd'2 = 354 |
A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their five previous scores which are:
|
X |
Y |
|
25 |
50 |
|
85 |
70 |
|
40 |
65 |
|
80 |
45 |
|
120 |
80 |
Which batsman should be selected if we want
(i) a higher run getter, or
(ii) a more reliable batsman in the team?
Batsman -A
|
Scores (X) |
|
|
|
25 |
–45 |
2025 |
|
85 |
+15 |
225 |
|
40 |
–30 |
900 |
|
80 |
+10 |
100 |
|
120 |
+50 |
2500 |
|
ΣX= 350 |
Σx2= 50 |
Calculate mean deviation from median from the following data:
|
X |
10 |
20 |
30 |
40 |
50 |
|
Y |
2 |
8 |
15 |
10 |
4 |
|
X |
f |
cf |
D |
fD |
|
10 |
2 |
2 |
20 |
40 |
|
20 |
8 |
10 |
10 |
80 |
|
30 |
15 |
25 |
0 |
0 |
|
40 |
10 |
35 |
10 |
100 |
|
50 |
4 |
39 |
20 |
80 |
|
Σf =39 |
ΣfD = 300 |
Calculate standard deviation from the following data:
|
S.No. |
1 |
2 |
3 |
4 |
5 |
|
Monthly |
400 |
600 |
900 |
1400 |
1200 |
|
Income (Rs.) |
|
SI. No. |
Monthly Income (Rs.) |
d |
d1 (d ÷ 100) |
d2 |
|
1 |
400 |
–500 |
–5 |
25 |
|
2 |
600 |
–300 |
–3 |
9 |
|
3 |
900 |
0 |
0 |
0 |
|
4 |
1400 |
500 |
5 |
25 |
|
5 |
1200 |
300 |
3 |
9 |
|
N = 5 |
AM = 900 |
Σ d1 = 0 |
Σd2 = 68 |
Calculate range and co-efficient of range from the following data:
4, 7, 8, 46, 53, 77, 8, 1, 5, 13
(i) Range= H-L = 77-l = 76
(ii) Co-efficient of Range
Find out quartile deviation and coefficient of quartile deviation of the following series.
Wages of 9 workers in Rs. are 170, 82, 110, 100, 150, 150, 200, 116, 250
Rearranging the wages, we get
82, 100, 110, 116, 150, 150, 170, 200, 250
How is dispersion of the series different from average of the series?
Average of the series refers to central tendency of series whereas dispersion measures the extent to which different items tend to disperse away from the central tendency.
Name the methods of absolute measures of dispersion.
(i) Range (ii) Quartile deviation, (iii) Mean deviation, (iv) Standard deviation, (v) Lorenz curve.
What is the principal drawback of mean deviation as a measure of dispersion?
The principal drawback of mean deviation dispersion is that all deviations from the avarage value of the series are taken as positive, even when some of these are actually negative.
Give one point of difference between mean deviation and standard deviation.
In the calculation of mean deviation, deviation may be taken from mean, median or mode, but in the calculation of standard deviations are taken only from the mean value of the series.
What is difference between coefficient of variation and variance?
Co-efficient of variation is estimated
as 
whereas variance is the square of standard deviation
How do range and quartile deviation measure the dispersion?
Range and quartile deviation measure the dispersion by calculating the spread within which the values lies.
What do mean deviation and standard deviation calculate?
Mean deviation and standard deviation calculate the extent to which the values differ from the average.
Name the measures which are based upon the spread of values.
(i) Range, (ii) Quartile deviation.
Give any two demerits of range.
(i) Range is unduly affected by extreme values.
(ii) It cannot be calculated for open-end distribution.
Name any two measures of dispersion from average.
(i) Mean deviation and (ii) Standard deviation.
When is the mean deviation the least and when is higher?
Mean deviation is the least when calculated from the median and will be higher if calculated from the mean.
Can standard deviation in case of individual series be calculated from the values directly i.e. without taking deviations while using the direct method? Write down the formula.
Yes. In this case following formula is used: 
Write down any two merits and two demerits of mean deviation.
Two merits of mean deviation:
1. Mean deviation is less affected by extreme values than the range.
2. It can be calculated from any average (mean, median, mode)
Two demerits of mean deviation:
1. It is not capable of any further algebraic treatment
2. Calculation of mean deviation suffers from inaccuracy because the ‘+’ or ‘–’ signs are ignored.
Semi-inter quartile range is also known as:
Mean deviation
Standard deviation
Quartile deviation
Quartile range
C.
Quartile deviation
M.D. represents:
Mean deviation
Median deviation
Marginal deviation
None of above
A.
Mean deviation
cf is used for:
Common factor
Cumulative frequency
Common value
None of above
B.
Cumulative frequency
Value of median is:
Second quartile
First quartile
Third quartile
Fourth quartile
A.
Second quartile
Which is the correct statement?
In the calculation of standard deviation, deviations may be taken from mean, median or mode.
In the calculation of standard deviations, signs of deviations (+) or (–) are ignored.
In the calculation of standard deviation, signs are not ignored.
In the calculation of mean deviation, deviations are taken only from the mean value of the series.
C.
In the calculation of standard deviation, signs are not ignored.
represents:
Lower quartile
First quartile
Both (a) and (b)
Neither (a) nor (b)
A.
Lower quartile
Decile is the division of the series into:
Two parts
Fifty parts
Ten parts
Hundred parts
B.
Fifty parts
Range is the:
Ratio of the largest to the smallest observations
Average of the largest to smallest observations
Difference between the smallest and the largest observations
Difference between the largest and the smallest observations
D.
Difference between the largest and the smallest observations
Write the false statement:
Range is the easiest and simplest measure because it is the difference between two extreme items
Quartile deviation is superior to the range as it is not affected too much by the value of extreme items
Range and quartile deviation are based on all the items of series
Range and quartile deviation are useful for general study of variability
C.
Range and quartile deviation are based on all the items of series
Which is not the relative measure of dispersion?
Co-efficient of range
Lorenz curve
Co-efficient of mean deviation
Co-efficient of variation
C.
Co-efficient of mean deviation
Sponsor Area
Sponsor Area
(X)
(X2)
d
(x)
2
d1
is used to calculate:
(x)
2
d1
d
(X)
(X2)
