Introduction to Measure of Dispersion

A modern student of statistics is mainly interested in the study of variability and uncertainty. In this section we shall discuss variability and its measures, and uncertainty will be discussed in probability.

We live in a changing world, and changes are taking place in all areas of life. The study of statistics does not show much interest in things which are constant. The total area of the Earth may not be very important to a research-minded, person but the area covered by different crops, forests, residential and commercial buildings are figures of great importance, because these figures keep on changing from time to time and from place to place. Many experts are engaged in the study of changing phenomena.

Experts working in different countries keep a watch on forces which are responsible for bringing changes in the fields of human interest. Agricultural, industrial and mineral production and their transportation from one area to other parts of the world are of great interest to economists, statisticians, and other experts. Changes in human populations, changes in standards of living, changes in literacy rates and changes in prices attract experts to perform detailed studies and then correlate these changes to human life. Thus variability or variation is connected with human life and its study is very important for mankind.

Dispersion

The word dispersion has a technical meaning in statistics. The average measures the center of the data, and it is one aspect of observation. Another feature of the observation is how the observations are spread about the center. The observations may be close to the center or they may be spread away from the center. If the observations are close to the center (usually the arithmetic mean or median), we say that dispersion, scatter or variation is small. If the observations are spread away from the center, we say dispersion is large.

Suppose we have three groups of students who have obtained the following marks on a test. The arithmetic means of the three groups are also given below:

Group A: 46, 48, 50, 52, 54       ${\overline X _A} = 50$
Group B: 30, 40, 50, 60, 70       ${\overline X _B} = 50$
Group C: 40, 50, 60, 70, 80       ${\overline X _C} = 60$

In groups A and B the arithmetic means are equal, i.e.${\overline X _A} = {\overline X _B} = 50$. But in group A the observations are concentrated around the center. All students in group A have almost the same level of performance. We say that there is consistency in the observations in group A. In group B the mean is 50 but the observations are not close to the center. One observation is as small as 30 and one observation is as large as 70. Thus there is greater dispersion in group B. In group C the mean is 60 but the spread of the observations with respect to the center 60 is the same as the spread of the observations in group B with respect to their own center, which is 50. Thus in groups B and C the means are different but their dispersion is the same. In groups A and C the means are different and their dispersions are also different. Dispersion is an important feature of observation and it is measured with the help of the measures of dispersion, scatter or variation. The word variability is also used for the idea of dispersion.

The study of dispersion is very important in statistical data. If in a certain factory there is consistency in the wages of workers, the workers will be satisfied. But if some workers have high wages and some have low wages, there will be unrest among the low paid workers and they might go on strike and arrange demonstrations. If in a certain country some people are very poor and some are very rich, we say there is economic disparity. This means that dispersion is large.

The idea of dispersion is important in the study of workers' wages, price of commodities, standards of living of different people, distribution of wealth, distribution of land among framers, and many other fields of life. Some brief definitions of dispersion are:

1. The degree to which numerical data tend to spread about an average value is called the dispersion or variation of the data.
2. Dispersion or variation may be defined as a statistic signifying the extent of the scatteredness of items around a measure of central tendency.
3. Dispersion or variation is the measurement of the size of the scatter of items in a series about the average.