Statistics For Economics Chapter 4 Presentation Of Data

A.

One-dimensional diagram

B.

mode

C.

median

A.

long term trend

(i) Bar Diagram.

(ii) Sub-divided or Component Bar Diagram.

(iii) Component Bar Diagram.

Increase in the share of urban non-workers:

Lower level of urbanisation (in percentage)

Alternatively : (Increase in the share of urban non-workers (for the state of Delhi) :

(in percentage)

(Note : arbitrary figures have been taken.)

Procedure of drawing a histogram differ when class-intervals are unequal in comparison to equal class-intervals in a frequency table. In the case of equal class-intervals, no adjustment is made. In this case, the width of the rectangles would be of equal width, whereas the length of the rectangles would be different in proportion to the frequencies of the class-intervals.

In case of unequal class-intervals, frequencies of unequal class-intervals are adjusted. For adjustment, following steps are taken into account:

1. We first locate the lowest class-interval. We make no adjustment in this class-interval.

2. We adjust the frequencies of all other classes with the lowest class-size using the following formula:

Adjusted frequency

3. The histogram is made on the basis of adjusted frequencies and not the frequencies given originally.

4. We below given the adjusted frequency for the further clarification. Class-Interval f Adjusted frequency

It may be pointed out that the width of the rectangles would be different

(i)

(ii) We will use multiple bar diagram.

Presentation of data means exhibition of the data in such a clear and attractive manner that these are easily understood and analysed.

A table is a systematic organisation of data in columns and rows.

1. General purpose table. 2. Special purpose table. 3. Original table. 4. Derived table. 5. Simple or one way table. 6. Complex table.

A simple table is that which shows only one characteristic of the data.

Tabulation refers to the method or process of presenting data in the form of rows and columns.

A complex table is one which shows more than one characteristics of the data.

1. Table Number, 2. Title and Head Note.

Bar diagrams are those diagrams in which data are presented in the form of bars or rectangles.

Multiple bar diagrams are those diagrams which show two or more sets of data simultaneously.

Pie diagram is a circle divided into various segments showing percent values of a series.

Sub-divided bar diagrams show total values as well as part values of a set of data.

Frequency diagrams those diagram which present frequen distribution diagramatically.

A frequency curve is a curve which is plotted by joining the mid-points of all tops of a histogram by free hand smoothed curves.

Ogive or Cumulative frequency curve is the curve which is constructed by plotting frequency data on the graph paper, in the form of smooth curve.

Frequency polygon is drawn by joining the mid-points of all tops of a histogram using a foot-rule (to make a straight line).

Graphs showing arithmetic values of variables are called time-series graphs.

(i) One variable graph. (ii) Two or more than two variable graphs.

Two merits of graphic presentation are : (i) Simple and understandable information, (ii) Lasting impact.

1. Limited use and 2. Misuse

These are types of classification.
(i) Quantitative i.e. in terms of magnitudes.

(ii) Chronological (or Temporal) i.e. on the basis of time.

(iii) Geographical (or Spatial) i.e. areawise.

(iv) Quantitative i.e. according to some attribute.

If the data are classified on the basis of same characteristics capable of quantitative measurement such as height, age, income, expenditure, marks scored by students in class etc., the classification is known as quantitative classification.

When the data are classified on the basis of time, it is known as chronological classification. Here, one variable is time.

When the data is classified on the basis of place, it is known as geographical classification.

If the data are classified on the basis of some attribute or quality (descriptive characteristic) such as sex, literacy, beauty, honesty, intelligence, education, etc., the classification is known as qualitative classification.

When on the basis of presence and absence of an attribute the data are classified into two classes, one possessing that attribute and the other not possessing that attribute it is called two-fold classification.

A quantity which can assume a range of numerical values is called a variable and each value within the range is called variate or observation.

Classification forms a basis for tabulation and is done prior to tabulation. Tabulation is systematic arrangement of data in column and rows.

That the advantages of tabulation are as under:

1. The tabulated data can be more easily understood.

2. The data becomes attractive and leaves a lasting impression.

Diagrams and graphs are pictorial representation of statistical data. They are also called diagrammatic representation of data.

Advantages of diagrammatic representation of data:

1. Diagrams give a very clear picture of data.

2. Comparison between different samples with the regard to certain statistical measures like mean, median and mode can be easily made without using any statistical technique.

Limitations of diagrammatic representation:

1. A limited set of data can be presented in the form of a diagram.

2. Diagram do not present the small differences properly.

Bar diagrams are one dimensional because their height represents the size of the figure but not the width.

In simple bar diagram equal width is given to all the lines to make them look as rectangles e.g. students, prices, etc.

Limitations of bar diagram:

1. They can be used only when the given figures do not vary much and the proportion between the figure is small.

2. This can be used to show only one variable.

Component or sub-divided diagram are useful for presenting several items of a variable graphically and enable us to make comparative study of different parts among themselves and study the relationship between each part and the whole.

Multiple bar diagrams are used when two or more sets of inter-related variables are to be presented graphically for comparison. A set of simple adjacent bars is drawn. Comparisons can be made either between two or more variable or magnitudes of one variables in two or more aspects.

Sub-divided or component bar diagram presented graphically on precentage basis give percentage bar diagram.

Pie diagram is a circle divided into component sectors with areas proportional to the size of the corresponding component. They are prepared on percentage basis. This diagram is used to compare the relationship between various components. For drawing the Pie-diagram, percentage of each sector is converted into degrees keeping in view that the whole circle covers 360°.

An histogram is a graphical representation of a frequency distribution of a continuous series. It represents the class frequencies in a frequency distribution by verticle rectangles meeting each other from left to right. Histograms are two dimensional figures in which both length and breadth of the rectangles are considered.

A frequency curve for a grouped frequency distribution is a smooth freehand curve drawn by joining the mid-points of the upper horizontal sides of the histogram drawn over that frequency distribution.

The basis of classification is time.

Example of qualitative classification :

Literacy in Bihar (in percentage)

Here the series is not continuous. Hence, we have to make it continuous for doing so, we will divide the difference of class boundaries by 2. The resultant figure i.e. 5 by be deducted from the lower limit and added in the upper limit. Hence:

Example of quantitative classification :

Example of spatial classification:

Exports by India to other countries in a particular year

Example of spatial classfication:

Exports by India to other countries

xample of temporal classification:

The sale proceeds of a tea shop during the Year 1995 - 2000

Following are the objectives (purposes) of classification:

1. To condense the mass of data in such a manner that similarities and dissimilarities are reedily apprehended and relationships studies.

2. To facilitate comparisons.

3. To have a bird’s eye-view of the significant feature of the data.

4. To englight important information while giving less prominence to insignificant items.

5. To utilise the data for tabulation and further statistical analysis.

6. To eliminate unnecessary details contained in the raw data.

7. To present the complex, scattered in a concise, logical and understable form.

Following are the objectives of tabulation:

1. In tabulation the data are presented systematically in columns and rows in a concise form. Thus, data becomes more meaningful and lot of time is saved in its study and understanding.

2. When the data are arranged in tables with titles and table number, they can be easily identified and made use of a source reference for future studies.

3. Tabulation of data shows the trend of the information under study and depicts the patterns with in the figures which cannot be understood in a descriptive form of presentation.

4. It is only after tabulation that data becomes fit for statistical processing.

ollowing are the advantages of diagrammatic representation :

1. Diagrams give a very clear picture of data.

2. Comparison between different samples with the regard to certain statistical measures like mean, median, mode etc. can be easily made without using any statistical technique.

3. Diagrams can be used universally at any place and at any time.

4. Diagrams can be used for numerical type of statistical analysis.

5. It saves time and energy and it is economical.

6. The data can be remembered easily.

1. Diagrammatic presentation of data is just an approximation of the actual behaviour of the variables.

2. Only a limited set of data can be presented in the form of a diagram.

3. Diagrammatic presentation of data is a time- consuming process.

4. It is not very easy to arrive at final conclusion after seeing the diagram.

5. Diagrams do not show small differences properly.

6. Diagrams can be analysed mentally and are not amendable to further statistical treatment at the tabular presentation. Diagrams are drawn on false baseline.

7. Diagrams can be used only for comparative study.

8. Diagrams are capable of being misused easily.

Sexwise distribution of Union and Non-union Members for 1960, 1965, 1970

Following are the steps involved in the preparation of Pie diagram:

1. Convert each component as a percentage of the total.

2. Multiply the percentage by 360/100 = 3.6 to convert into degrees.

3. Starting with the twelve o’clock position on the circle (clockwise) draw the largest component circle.

4. Draw other components in clockwise succession in descending order of magnitude except for each-all components like all others and miscellaneous which are shown last.

5. Use different columns or shades to distinguish between different components.

6. Explain briefly the different components either within the components in the figure or outside by arrows.

Difference between frequency polygon and histogram:

1. Frequency polygon is an improvement over histogram because it provides a continuous curve indicating the causes of rise and fall in the data. On the other hand, frequency polygon is an approximate curve, but still it is more usefui as compared to histogram.

2. In the frequency polygon, it is assumed that the frequency distribution in a particular class-width whereas histogram may be used to represent frequency distribution with equal as well as with unequal class width.

3. In case of frequency polygon, it is assumed that all frequencies in a particular class are concerned at the mid point of that class whereas in case of histogram, it is supposed that they are evenly spread over the class interval.

When class intervals are equal i.e. when all rectangles have the same base, area can conveniently be represented by the frequency of any interval for purposes of comparison. When bases vary in their width, the heights of rectangles are to be adjusted to yield comparable measurements. In such a situation frequency density (class frequency divided by width of the class interval) instead of absolute frequency.

Distinction between percentage subdivided bar diagram and subdivided bar diagram are as follow :

ollowing are the rules of constructing diagrams:

1. Every diagram should be suitably and briefly titled. It can be mentioned either at the top of the diagram or below it.

2. It should suit to the size of the paper.

3. It should be neat and attractive.

4. It should be clearly indexed.

5. Diagrams should contain footnotes and a proper spacing between the figure.

6. The details in the diagrams should be self explanatory.

Following are the steps involved in the preparation of percentage sub-divided bar diagram:

1. Convert the data into percentage form by dividing 100.

2. Cumulative frequency is obtained.

3. Separate bars are prepared for different years. But all the bars have same length or of 100.

4. Then as per cumulative frequency bars are sub-divided into different categories shown by different marks expressed in percentages.

General Purpose Table has no special importance as it is used by different persons in a different manner. Generally, it is given on the back of reports or circulars etc.

Special Purpose Table are given for specific purposes. They serve the purpose of that particular group for which they have been prepared. These are brief in nature and are targeted towards a particular objective.

Before making the histogram, we have to convert the class intervals :

The formation of discrete frequency distribution is quite simple. The number of times a particular value is repeated is noted down and mentioned against that values instead of writing that value repeatedly, e.g. in the distribution 3, 4, 6, 6, 8, 8, 8, 8, 1, 1, 1, we can write:

Classes can be formed in two ways :

(i) Exclusive type, (ii) Inclusive type.

(i) Exclusive Type : When the class intervals are so fixed that the upper limit of one class is the lower limit of the new class, it is known as exclusive method of clasfication.

For example:

In this method, higher value of the variable in the class is not included in that class. i.e.

(ii) Inclusive Type : In this method, the students getting say 39% marks will be included in the class 30 - 39 itself i.e.

Multiple Bar diagram will serve our purpose:

Here we will take the help of subdivided bar diagram to represent the above data: 

This is a case of unequal class intervals.

Difference between ‘less than’ and ‘more than’ ogives:

The total area under a frequency curve represents the total frequencies.

Both frequency polygon and frequency curve are drawn by joining the mid-points of all tops of a histogram. But in case of frequency polygon, the points are joined using a foot rule to make a straight line, but in case of frequency curve, the points are joined using free hand.

A histogram becomes a frequency polygon when we draw a line joining mid-points of the tops of all rectangles in a histogram by using a first rule.

Frequencies are comulated.

Values are arrayed.

Conversion of Inclusive series into Exclusive series:

Method of conversion of inclusive series into exclusive series :

1. First find the difference between the upper limit of a class interval and the lower limit of the next class interval.

2. Half the difference is added to the upper limit of each class interval and remaining half is deducted from the lower limit of each class-interval. In the question the upper limit of the first class interval is 49 and the lower limit of the second class interval is 50. The difference between them is one and its half is 0.5, 0.5 will be subtracted from 40, and 0.5 will be added to 49. Hence, first class interval will be 39.5 – 49.5.

The series will be cumulative, if the words- less than, more than, above, below, over, under, upto, exceeding, not exceeding etc, are given before all limits of the class-intervals.

1. Expenditure on food

2. Expenditure on clothing

For constructing histogram of unequal class-intervals, first we note a class of the smallest intervals. Other classes are noted in the increasing order of their intervals. If the size of one class-interval is twice the smallest size in the series, frequency of that class is divided by two. Suppose the class with the smallest interval is 5 – 10 and the class with the largest interval is 10 – 20, the frequency of which is 12. Here the class interval of the bigger class is 10 which is twice as much as the size of the class-interval of the smallest class i.e. 5. The bigger class interval is divided into two parts 10 – 15 and 15 – 20 and accordingly the frequency of the bigger class, 12 would be divided by 2 i.e. 12+ 6. In this way there will be following adjustment.

Foot-notes are used to explain the complex nature of a table.

At the end of the table on the right side.

Ogive curve.

Adjustment factor for any class =  Class interval of the concerned class
                                                         Lower class interval

Adjustment factor = 10/5 = 2