# 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. Changes are taking place in every sphere of life. A man of statistics does not show much interest in those things which are constant. The total area of the earth may not be very important to a research minded person but the area under different crops, area covered by forests, area covered by residential and commercial buildings are figures of great importance because these figures keep on changing form time to time and from place to place. Very large number of experts is engaged in the study of changing phenomenon. Experts working in different countries of the world keep a watch on forces which are responsible for bringing changes in the fields of human interest. The agricultural, industrial and mineral production and their transportation from one part to the other parts of the world are the matters of great interest to the economists, statisticians, and other experts. The changes in human population, the changes in standard living, and changes in literacy rate and the changes in price attract the experts to make detailed studies about them and then correlate these changes with the human life. Thus variability or variation is something connected with human life and study is very important for mankind.

Dispersion:

The word dispersion has a technical meaning in statistics. The average measures the center of the data. It is one aspect observations. Another feature of the observations is as to how the observations are spread about the center. The observation may be close to the center or they may be spread away from the center. If the observation 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 in 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 a group A and B arithmetic means are equal i.e.${\overline X _A} = {\overline X _B} = 50$. But in group A the observations are concentrated on the center. All students of group A have almost the same level of performance. We say that there is consistence in the observations in group A. In group B the mean is 50 but the observations are not closed 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 group B and C the means are different but their dispersion is the same. In group A and C the means are different and their dispersions are also different. Dispersion is an important feature of the observations and it is measured with the help of the measures of dispersion, scatter or variation. The word variability is also used for this idea of dispersion.

The study of dispersion is very important in statistical data. If in a certain factory there is consistence 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 strikes and arrange demonstrations. If in a certain country some people are very poor and some are very high rich, we say there is economic disparity. It means that dispersion is large. The idea of dispersion is important in the study of wages of workers, prices of commodities, standard of living of different people, distribution of wealth, distribution of land among framers and various 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 statistics signifying the extent of the scatteredness of items around a measure of central tendency.
3. Dispersion or variation is the measurement of the scatter of the size of the items of a series about the average.