Statistics For Economics Chapter 3 Organisation Of Data

Example : Marks obtained by 10 students in Class XI are : 38 45 42 36 30 40 60 65 70 11

(a) Series of invidividual observations.

(b) Discrete series.

(c) Continuous series.

Data may be arranged by time, or space or both. For example, we have time series data on aggregate income, aggregate consumption, size of the population, etc.

Continuous Variable : It can take all values in a given range. For example, heights and weights of individuals, prices of commodities, income of individuals may be treated as continuous variables.

Discrete Variables : If the variables can take only some particular value like whole numbers, it is called a discrete variable. For example, number of students in different classes or different schools or the size of the households.

Bivariate Frequency Distribution : A bivariate frequency distribution is the frequency distribution of two variables. For example the table shows the sales and advertisement expenditure of the firm. They are classed in different rows.

1. It helps in the comparison of data.

2. It helps us to understand the relationship among variables.

3. It highlights significant features of the data at a glance.

4. It makes the statistical analysis of the data easier.

5. It arranges and presents huge volume of raw data in meaningful and condensed form.

However, in Statistics, the data on any single variable, or a set of variables, for all individual units in a region, constitute the population of that variable, or variables.

Univariate : If the data are on a single variable, the set of measurements constitutes a univariate population of that variable. We have a bivariate or multivariate population of the set variables. For example, we may have a univariate population of prices, or a population of incomes. We may have a bivariate population of height and weights of all individuals in a region.
Multivariate : A multivariate population of expenditures on various items of consumption of all households.

2. in case of exclusive series, value of the upper limit of a class is included in the lower limit of the next class interval. In inclusive series value of the upper limit of a class is included in that very class interval.

3. Exclusive series is useful whether the value is complete number or in decimals, but inclusive series is useful only when value is in complete number.

Exclusive Method : This method is useful whether the value is complete number or in decimals. In case of exclusive series, value of the upper limit of a class is included in the lower limit of the next class interval. Value of the upper limit of the class is included in that very class interval.

Inclusive Series : In inclusive series value of the upper limit of a class is included in that very class interval. Inclusive series is useful when value is incomplete number.

In order to get a better idea about the distribution of values within the range, we should subdivide the total range into a number of class intervals and find out the number of values in different classes.

The main points underlying the construction of a frequency distribution are as follows :

(a) Construction of Discrete Frequency Distribution :

1. Prepare a table with three columns-first for variable under study, second for 'Tally bars' 

and the third for the total, representing corresponding frequency to each value or size of the variable.

2. Place all the values of the variables in the first column in ascending order-beginning with the lowest end giving to the highest. The gap between one magnitude to another may preferably be the same.

3. Put pars (vertical lines) in front of the values according in the second column keeping in view the number of items a particular value repeats itself. This column is for facility in counting. Blocks of five bars orare prepared and some space is left between each block of bars.

4. Count the number of bars in respect of each value in the variable and place it in the third column made for total of frequency.

(b) Construction of Continuous Frequency Distribution :

There are two methods of classifying the data according to class intervals :

Exclusive Method : Under this method upper limits are excluded. The upper limits of class intervals is the lower limit of the next class. For example, if the students obtained marks are grouped as 5 – 10,10 – 15,15 – 20,20 – 25, 25 – 30 etc., we include in first group of students whose marks are 5 or more but under 10. If the marks of a students are 10 he is not included in the first group but in the second, i.e., 10 to 15.

Inclusive Method : Under this method upper class limits of classes are included in respective classes. For example, if the students obtained marks are grouped as 5 – 9, 10 – 14, 15 – 19,20 – 24,25 – 29 etc., in the group 5 – 9, we include in first group students whose marks is between 5 and 9. If the marks of students are 10 he is included in the next class, i.e., 10 to 14. If there are no whole numbers, the classes can be made 5 – 9.9, 10 – 14.9, 15 – 19.9 and so on.

1. The classical data become comparable.

2. Homogeneous data are kept or classified in one group.

(i) Homogeneity, (ii) Clarity, (iii) Stability, (iv) Flexibility, (v) Diversification.

A.

B.

A.

A.

A.

A.

B.

A.

A.

A.

The average of the upper class limit and the lower class limit.

B.

 Bivariate Distribution

A.

the actual values of observations

A.

The upper class limit of a class is excluded in the class interval

A.

difference between the largest and the smallest observations.

Advantage of Classification:

1. It helps in the comparison of data.

2. It helps us to understand the relationship among variables.

3. It highlights significant features of the data at a glance.

4. It makes the statistical analysis of the data easier.

5. It arranges and presents huge volume of raw data in meaningful and condensed form.

n statistics the term variable is used, only if the changing characteristics can be numerically measured. The heights and weights of individuals are variables, as they can be measured in numerical terms. Prices of commodities vary time and space, and they can be numerically measured. So the price is variable.

Continuous Variable : It can take all values in a given range. For example, heights and weights of individuals, prices of commodities, income of individuals may be treated as continuous variables.

Discrete VariablesIf the variables can take only some particular value like whole numbers, it is called a discrete variable. For example, number of students in different classes or different schools or the size of the households.

There are two methods of classifying the data according to class-intervals, namely Exclusive Method and Inclusive Method.

Exclusive Method : This method is useful whether the value is complete number or in decimals. In case of exclusive series, value of the upper limit of a class is included in the lower limit of the next class interval. Value of the upper limit of the class is included in that very class interval.

Inclusive Series : In inclusive series value of the upper limit of a class is included in that very class interval. Inclusive series is useful when value is incomplete number.

(i) Range = Largest value-Smallest value = 5500 - 1007 = 4083.

(ii).

(iii) (a) Number of household whose expenditure on food is less than Rs. 2000 = 19 + 14 = 33

(b) More than Rs. 3000 = 2+1 + 2+1 = 6

(c) Between Rs. 1500 and Rs. 2500 = 14 + 6 = 20.

The classification of data as a frequency distribution has an inherent shortcoming. While it summarises the raw data making it concise and comprehensible, it does not show the details that are found in raw data. There is a loss of information in classifying raw data though much is gained by summarising it as a classified data. Once the data are grouped into classes, an individual observation has no significance in further statistical calculations. For example : the class 20–30 contains 6 obervations : 25, 25, 20, 22, 25 and 28. So when these data are grouped as a class 20–30 in the frequency distribution,the latter provides only the number of records in that class (i.e. frequency = 6) but not there actual values. All values in this class are assumed to be equal to the middle value of teh class interval or class mark (i.e. 25). Further statistical calculations are based only on the values of class mark and not on the values of teh observations in that class. This is true for other classes as well. Thus the use of class mark instead of the actual values of the obervations in statistical methods involves considerable loss of information.

The data collected from primary and secondary sources are raw data or unclassified data. Once the data is collected, the next step is to classify them for further statistical analysis. Classification brings order in the data. It is a tedious task to get information from large unclassified data. The raw data are summarised and made comprehensible by classification. When facts of similar characteristics are placed in the same class, it enables one to locate them easily, make comparison and draw inferences without any difficulty.

Univariate Frequency Distribution : The frequency distribution of a single variable is called a univariate distribution. For example marks of a student.

Bivariate Frequency Distribution : A bivariate frequency distribution is the frequency distribution of two variables. For example the table shows the sales and advertisement expenditure of the firm. They are classed in different rows.

Variable is a characteristic which is capable of being measured and changes its value over time. A variable may be either discrete or continuous.

Frequency Distribution

Classification is the grouping of related facts into different classes.

Statistical Data classified into classes or groups on the basis of their numerical values is called quantitative classification.

Arranging of data in different classes according to a given order is called series.

The extreme values of a class are limits.

Exclusive series is that series in which every class interval excludes items corresponding to its upper limit.

An inclusive series is that series which include all items upto its upper limit.

Magnitude of a class interval is the difference between the upper limit and the lower limit of a class.

Frequency is the number of times an item occurs (or repeat itself) in the series.

Individual series are those series in which the items are listed singly.

A discrete series or freqnency array is that series in which data are presented in a way that exact measurement of items are clearly shown.

It is that series in which items cannot be exactly measured. The items assume a range of values and are placed within the range or limits.

An open ended series is that series in which lower limit of the first class interval and the upper limit of last class interval is missing.

Cumulative frequency series is that series in which the frequencies are continuously added corresponding to each class interval in the series.

Magnitude = 55 – 15 = 40.

2, 4, 5, 8, 14, 25.

32, 22, 17, 15, 13.

The data in their original form are called raw data.

Example : Marks obtained by 10 students in Class XI are :

38 45 42 36 30 40 60 65 70 11

The term frequency signifies the number of items or a value repeated in a given set of information.

When individual data are arranged in an increasing or decreasing order, it is known as an array.

A series is a systematic arrangement of items in a particular order or sequence.

These are three types of series as under:

(a) Series of invidividual observations.

(b) Discrete series.

(c) Continuous series.

A series is called discrete when the frequency of various items are given separately

The presentation of observations of classes horizontally and individual items which fall in class are written according to ascending order of magnitude.

The type of presentation where the frequencies are represented by adding the frequency of each previous class.

Classification done according to quantitative varieties like marks, wages, etc. is termed as quantitative classification.

Classification according to attributes like honesty, beauty is known as qualitative classification.

‘Classification’ means arranging things in an appropriate order and putting them into homogeneous groups. For example, in library, the books and periodicals are classified and arranged according to subjects; students are grouped according to division they secure in a certain examination; plants and animals may be assigned to groups distinguished by structure, origin, etc.

Data may be arranged by time, or space or both. For example, we have time series data on aggregate income, aggregate consumption, size of the population, etc.

In Ascending Order

In Descending Order

Variable : In Statistics, the term ‘variable’ is used, only if, the changing characteristics can be numerically measured. Thus, heights and weights of individuals are variables, as they can be measured in numerical terms, price of commodities vary over time and space and they can be numerically measured. Therefore prices is a variable. Similarly, incomes of individuals, household expenditures on various items of consumption, size of households, input and outputs of firms are all variables.

Attribute : The looks of people, their intelligence and aptitude for art and music change from one individual to the other, they cannot be measured numerically in the same way as heights and weights, or prices and incomes. Therefore, they are not called variable in the statistical sense. They are called ‘attributes’. We may rank individual according to the quality of attributes. The ranks are sometimes used as their numerical values for purposes of analysis.

Continuous Variables : It can take all conceivable values in the given range. For example, heights and weights of individuals, prices of commodities, income of individuals may be treated as continuous variables. Although, in practice, the measurements are taken only approximately, upto one or two places of decimal, the true values may be anything in a certain range.

Discrete Variables : If the variables can taken only some particular value (like whole numbers), it is called a ‘discrete’ or ‘discontinuous’ variable. For example, number of students in different classes, different schools, or the size of households are discrete variables, as they can take interval values only.

In common language the word ‘population’ means the number of persons living in a certain region. We may count the number of persons and obtain the size of population of that region. Similarly, we may find the population of certain animals in forests in a country or the population of certain plants in a garden and so on. The term population implies ‘head count’.

However, in Statistics, the data on any single variable, or a set of variables, for all individual units in a region, constitute the population of that variable, or variables.

Univariate : If the data are on a single variable, the set of measurements constitutes a univariate population of that variable. We have a bivariate or multivariate population of the set variables. For example, we may have a univariate population of prices, or a population of incomes. We may have a bivariate population of height and weights of all individuals in a region.

MultivariateA multivariate population of expenditures on various items of consumption of all households.

After the data has been collected either with the help of the sampling method or census method the work of its organisation starts. The organisation of data means a systematic arrangements of collected figures so that the data becomes easy to understand and more convenient for further statistical treatment. This systematic arrangement changes into a statistical series.

1. In case of exclusive series, the upper limit of one class interval is the lower limit of the next class interval.

2. in case of exclusive series, value of the upper limit of a class is included in the lower limit of the next class interval. In inclusive series value of the upper limit of a class is included in that very class interval.

3. Exclusive series is useful whether the value is complete number or in decimals, but inclusive series is useful only when value is in complete number.

India’s population in the year 1951 was the minimum. It was 35.7 crores. Maximum population was in the year 2001. It was 102.7 crores.

(i) Maximum concentration of data is in class (50 – 60).

(ii) Minimum concentration of data is in class (0 – 96; 10).

he largest value of x is B and smaller value is A. Then x = B – A is the total range of x. A large range indicates that the values of x are spread over a large interval or the variation of values of x is large. A small range indicates smaller variation in the values of x. Thus, the range is measures of variation (or dispersion) of x.

In order to get a better idea about the distribution of values within the range, we should subdivide the total range into a number of class intervals and find out the number of values in different classes.

The main points underlying the construction of a frequency distribution are as follows :

(a) Construction of Discrete Frequency Distribution :

1. Prepare a table with three columns-first for variable under study, second for ‘Tally bars’ and the third for the total, representing corresponding frequency to each value or size of the variable.

2. Place all the values of the variables in the first column in ascending order-beginning with the lowest end giving to the highest. The gap between one magnitude to another may preferably be the same.

3. Put bars (vertical lines) in front of the values according in the second column keeping in view the number of items a particular value repeats itself. This column is for facility in counting. Blocks of five bars or are prepared and some space is left between each block of bars.

4. Count the number of bars in respect of each value in the variable and place it in the third column made for total of frequency.

(b) Construction of Continuous Frequency Distribution :

There are two methods of classifying the data according to class intervals :

Exclusive Method : Under this method upper limits are excluded. The upper limits of class intervals is the lower limit of the next class. For example, if the students obtained marks are grouped as 5 – 10,10 – 15,15 – 20,20 – 25, 25 – 30 etc., we include in first group of students whose marks are 5 or more but under 10. If the marks of a students are 10 he is not included in the first group but in the second, i.e., 10 to 15.

Inclusive Method : Under this method upper class limits of classes are included in respective classes. For example, if the students obtained marks are grouped as 5 – 9, 10 – 14, 15 – 19,20 – 24,25 – 29 etc., in the group 5 – 9, we include in first group students whose marks is between 5 and 9. If the marks of students are 10 he is included in the next class, i.e., 10 to 14. If there are no whole numbers, the classes can be made 5 – 9.9, 10 – 14.9, 15 – 19.9 and so on.

There is no hard and fast rule about how many class we choose; but as a working rule the number of classes should lie between 5 and 15. It should be noted that the number of classes will be large if we choose small size class intervals and it will be small if the size of class intervals is large.

As an illustration, suppose the range is 70, and we choose classes of width 2 each. We would require 70 ÷ 2 = 35 classes. However, the number of classes would be 14 if the width of each class was 5.

Size of Class Intervals : We may choose all classes of the same width or of different width. In the case of equal class intervals the size of the class interval is determined as soon as we have decided about the number of classes.

Suppose n is the number of classes and all classes are of width h, then n × h = R.

Knowing the range R and number of classes

n we can abtain h = R/n as the width of class interval. If the range is 70 and we choose 10 classes, the width is 7.

Choice of Class Limits : Suppose x is a continuous variable, such that it can take any value in a given range. In that case, it is possible to choose class limits which are not equal to any of the observed values, For example, height of individuals is a continuous variable, even though, in practice, one can measure height to the nearest of the unit value (in centimetres) as 165, 170, 169, 171 .........; or to the nearest of tenth place of decimals as 165.3, 170.4, 168.9, 170.8, ........ We may specify class intervals as 160.55 165.55,165.55 ....... so that none of the observed values of x is equal to any of the class limits.

Frequency Array : We obtain a frequency array if the variable x is discrete and we have frequencies corresponding to each value (there are no class intervals). Let us illustrate with the following example.

Example : A survey of 100 households was carried out to obtain information on their size, i.e., the number of members of households. The results of the survey are classified as a frequency array in table below :

Frequency Array of Size of Households

The column (1) of the table gives the values which the variable x (size of the households) takes; and column (2) gives the corresponding frequencies (number of households). Thus, there are 5 households whose size is 1, there are 15 households of size 2, and so on.

Frequency Distribution : The largest value of X is B and smallest value is A. Then X = B – A is the total range of X. A large range indicates that the values of X are spread over a large interval or the variation of value of X is large. A small range indicates smaller variation in the values of X. Thus, the range is measure of variation (or dispersion) of X.

For example : Suppose we have data on monthly income of 10,000 individuals, the maximum of which is Rs.50,000 and minimum is Rs. 1,000. Thus, the range is Rs.49,000. We observe that majority of individuals say, 70% have small incomes close to Rs. 5,000 and minority, say 2% have income close to Rs.30,000.

In order to get a better idea about the distribution of values within the range, we should subdivide the total range into a number of class intervals and find out the number of values in different classes.

Individual series : Under this method, the value of all the units are shown separately The following example will illustrate this:

Example : The marks obtained by 10 students in statistics are following :

The individual series may be arranged in following two orders :

(a) Ascending Order : When data are arranged in ascending order i.e., a small value to a big value it is known as arranging them in ascending order. The figures of above example may be arranged in ascending order as follows :

(b) Descending Order : When data are arranged serially starting from a big value to small value it is known as arrangement of data in descending order.

In these series all the items are divided in certain groups, but these groups are not continuous, therefore these series are known as discrete series. The numbered item that fall in every group are shown in each group which are known as frequencies. The following examples will illustrate this :

Examples :

(i) Discrete Series in Ascending Order :

(ii) Discrete Series in Descending Order :

Under such series all the variables are divided in certain continuous groups and their respective frequencies will be written with them. The following example will clear the form of such series:

Example:

Following are the elements of a continuous series:

(i) Class Intervals : These are the measurements in which some problems is measured and written in continuous group. In the above example, (0 – 5), (5 – 10) etc. are the class intervals of the series.

(ii) Limits of Class Intervals : Each class interval figure is known as limits of class interval. Small figures class intervals are known as lower limit class interval. In class interval (0 – 5) 0 is lower limit and 5 is a upper limit of this class interval.

(iii) Magnitude of Class Intervals The difference between upper limit and lower limit of a class interval is known as its magnitude. In class interval (0 – 5) 5 is the magnitude.

(iv) Mid Value The average of two limits of the class interval is known as mid value e.g., the mid value of class interval

(v) Frequencies : Number of repetition of items of various class intervals in the universe is known as frequencies which will be written with them.

Exclusive and Inclusive Continuous Series:

(a) Exclusive Series : Where the value of upper limit is not included in the same group, but will be included in next group, it is known as exclusive series e.g.

In the above series, 10, 20, 30,40, 50, and 60 will not be included in first, second third, fourth, fifth and sixth group respectively.

(b) Inclusive Series : Where value of upper limit is included in the same group, it is known as inclusive series e.g.,

bsolute errors : Absolute error is the difference between the true and estimated value.

Absolute error = actual value – estimated value

Relative errors : It is the ratio of absolute error to the estimated value. It is found out by dividing the absolute error by the estimated error.

Relative error

= Actual value - Estimated value
             Estimated error

Example : If the actual value is 400 and estimated value is 350. Find absolute and relative errors.

Solution : Here the actual value is 400 and estimated value is 350.

Absolute error = actual error – estimated value.

= 400 – 350 = 50

Relative error = 50 / 350 = 0.14

The relative error can also be expressed as percentage of 100.

In some series the lower class limit of the first class interval and the upper limit of the last class interval are missing. 'Less than' or below is specified in place of the lower class limit of the first class interval and 'More than' or above is specified in place of the upper class limit of the last class interval.

Example:

Cumulative frequency series is that series in which the frequencies are continuously added corresponding to each class interval in the series. The frequencies than become cumulative frequencies. The cumulative frequency for the first class interval is the same as frequency itself. But for the second class interval the cumulative frequency would be the both the second as well as first class intervals e.g.