Introduction

If we analyse two or more observations the central value may be the same but still there can be wide disparities in the formation of the distribution. For example, the AM of 2, 5 and 8 is 5; AM of 4, 5 and 6 is 5; AM of 1, 2 and 12 is 5; AM of 0, 1 and 14 is 5. Measures of dispersion will help us in understanding the important characteristics of a distribution.

This is explained with the help of another example.

Runs scored by three batsmen in a series of 5 one day matches are as given below:

Table 6.1 Cricket Scores
Days Batsman 1 Batsman 2 Batsman 3
1 100 70 0
2 100 80 0
3 100 100 300
4 100 120 180
5 100 130 20
Total 500 500 500
Mean 100 100 100

Since the average is the same in all the three cases, one Is likely to conclude that these three batsmen are alike, But a close examination shall reveal that the distributions differ widely from one another. In case of the first batsman, each and every item is perfectly represented by the AM and there is no dispersion. In case of the second batsman, only one item is perfectly represented by the AM and the other items vary, but the variation is not too much. In case of the third batsman, not a single item is represented by the AM. All the items vary, and the variation is too large. Here we can see that the first batsman is consistent, while the third is inconsistent.

Now it is quite obvious that averages try to tell only the representative size of a distribution. To understand it better, we need to know the spread of various items also. So in order to express the data correctly, it becomes necessary to describe the deviation of the observations from the central value. This deviation of items-from the central value is called dispersion.

" The degree to which numerical data tend to spread about an average value is called the variation or dispersion of the data." - Spiegel

The word dispersion means deviation or difference. In statistics, dispersion refers to deviation of various items of the series from its central value. Dispersion is the degree to which a numerical data tend to spread about an average value. Measure of dispersion is the method of measuring the dispersion or deviation of the different values from a designated value of the series. These measure, are also called averages of second order as they are averages of deviation taken from an average.

Objects of measuring variation

Measures of dispersion are useful in following respects:

  1. To test the reliability of an average: Measures of dispersion enable us to know whether an average is really representative of the series. If the variability in the values of various items in a series is large the average is not so typical. On the other hand, if the variability is small, the average would be a representative value.
  2. To serve as a basis for the control of the variability: A study of dispersion helps in identifying the causes of variability and in taking remedial measures.
  3. To compare the variability of two or more series: We can compare the variability of two or more series by calculating relative measures of dispersion. The higher the degree of variability the lesser is the consistency or uniformity and vice versa.
  4. To serve as a basis for further statistical analysis: Many powerful analytical tools in statistics such as correlation, regression, testing of hypothesis, analysis of fluctuations in time series, techniques of production control, cost control, etc., are based on measures of dispersion.

Methods of studying Dispersion

The following are the important methods:

  1. Range
  2. Quartile Deviation
  3. Mean Deviation
  4. Standard Deviation
  5. Lorenz Curve
Range and quartile deviation measure the dispersion by calculating the spread within which the values lie. Mean deviation and standard deviation calculate the extent to which the values differ from the average. Lorenz curve is a graphical method of finding dispersion.

Absolute and Relative Measures of Dispersion

Absolute measures of dispersion are expressed in the same statistical unit in which the original data are given. In case two sets of data are expressed in different units, absolute measures of dispersion are not comparable. In such cases, relative measures are used.

A measure of relative dispersion is the ratio of measure of absolute dispersion to an appropriate average. It is also called coefficient of dispersion, as it is independent of the unit.