Measures of Dispersion, Objectives, Meaning of Dispersion, Definition of Various Measures of Dispersion, Range, Mean Deviation from Mean, Variance, Standard Deviation

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Objectives:

After studying this lesson, you will be able to:

Explain the meaning of dispersion through examples;

Define various measures of dispersion – range, mean deviation, variance and standard deviation;

Calculate mean deviation from the mean of raw and grouped data;

Calculate mean deviation from the median of raw and grouped data.

Calculate variance and standard deviation of raw and grouped data; and

Illustrate the properties of variance and standard deviation.

Analyses the frequency distributions with equal means.

Meaning of Dispersion:

To explain the meaning of dispersion, let us consider an example.

Two sections of students each in class X in a certain school were given a common test in Mathematics (maximum marks ). The scores of the students are given below:

Section A:

Section B:

The average score in section A is .

The average score in section B is .

Let us construct a dot diagram, on the same scale for section A and section B (see Fig.)

The position of mean is marked by an arrow in the dot diagram.

The same scale for section A and section B

The Same Scale for Section a and Section B

The same scale for section A and section B

Clearly, the extent of spread or dispersion of the data is different in section A from that of B. The measurement of the scatter of the given data about the average is said to be a measure of dispersion or scatter.

In this lesson, you will read about the following measures of dispersion:

Range

Mean deviation from mean

Mean deviation from median

Variance

Standard deviation

Definition of Various Measures of Dispersion:

Range:

In the above cited example, we observe that

The scores of all the students in section A are ranging from to ;

The scores of the students in section B are ranging from to .

The difference between the largest and the smallest scores in section A is

The difference between the largest and smallest scores in section B is .

Thus, the difference between the largest and the smallest value of a data, is termed as the range of the distribution.

Mean Deviation from Mean:

Two sections of students each in class X in a certain school were given a common test in Mathematics (maximum marks ). The scores of the students are given below:

Section A:

Section B:

Let us now find the sum of deviations from the mean, i.e., for scores in section B.

Let Us Now Find the Sum of Deviations
Let us now find the sum of deviations

Observations

Deviations from mean

Again, the sum is zero. Certainly it is not a coincidence. In fact, we have proved earlier that the sum of the deviations taken from the mean is always zero for any set of data. Why is the sum always zero?

On close examination, we find that the signs of some deviations are positive and of some other deviations are negative. Perhaps, this is what makes their sum always zero. In both the cases, we get sum of deviations to be zero, so, we cannot draw any conclusion from the sum of deviations. But this can be avoided if we take only the absolute value of the deviations and then take their sum.

If we follow this method, we will obtain a measure (descriptor) called the mean deviation from the mean.

The mean deviation is the sum of the absolute values of the deviations from the mean divided by the number of items, (i.e., the sum of the frequencies).

Variance:

In the above case, we took the absolute value of the deviations taken from mean to get rid of the negative sign of the deviations. Another method is to square the deviations.

Let us, therefore, square the deviations from the mean and then take their sum. If we divide this sum by the number of observations (i.e., the sum of the frequencies), we obtain the average of deviations, which is called variance. Variance is usually denoted by .

Standard Deviation:

If we take the positive square root of the variance, we obtain the root mean square deviation or simply called standard deviation and is denoted by.

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