SSC CGL JSO Study Materials Day 5

SSC CGL JSO Study Materials

Common Measure of Dispersion

Introduction

Measure of dispersion has two terms, measure and dispersion.

™ ‘Measure’ here means specific method of estimation while ‘dispersion’ term means deviation or difference or spread of certain values from their central value.

You have learnt various measures of central tendency. Measures of central tendency help us to represent the entire mass of the data by a single value.

Can the central tendency describe the data fully and adequately?

In order to understand it, let us consider an example.

The daily incomes of the workers in two factories are:

Factory A:  35    45     50     65      70       90       100

Factory B:  60    65     65    65       65       65        70

Here we observe that in both the groups the mean of the data is the same, namely, 65

(i) In group A, the observations are much more scattered from the mean.

(ii) In group B, almost all the observations are concentrated around the mean.

Certainly, the two groups differ even though they have the same mean. Thus, there arises a need to differentiate between the groups. We need some other measures which concern with the measure of scatteredness (or spread). To do this, we study what is known as measures of dispersion.

Meaning of Dispersion

To explain the meaning of dispersion, let us consider an example. Two sections of 10 students each in class X in a certain school were given a common test in Mathematics (40 maximum marks). The scores of the students are given below:

Section A:  6     9    11    13    15    21     23      28      29       35

Section B: 15   16   16    17    18    19     20      21      23       25

The average score in section A is 19.

The average score in section B is 19.

Let us construct a dot diagram, on the same scale for section A and section B (see Fig. 29.1)

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

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.

Now, you will read about the following measures of dispersion:

(a) Range

(b) Quartiles deviation

(c) Mean deviation

(a) Range

Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution. Thus,

R = L – S

Higher value of Range implies higher dispersion and vice-versa.

Notwithstanding some limitations, Range is understood and used frequently because of its simplicity. For example, we see the maximum and minimum temperatures of different cities almost daily on our TV screens and form judgments about the temperature variations in them.

(b) Quartile Deviation

The presence of even one extremely high or low value in a distribution can reduce the utility of range as a measure of dispersion. Thus, you may need a measure which is not unduly affected by the outliers.

In such a situation, if the entire data is divided into four equal parts, each containing 25% of the values, we get the values of Quartiles and Median.

The upper and lower quartiles (Q₃ and Q₁, respectively) are used to calculate Inter Quartile Range which is Q₃ - Q₁

Inter- Quartile Range is based upon middle 50% of the values in a distribution and is, therefore,  not affected by extreme values. Half of the Inter-Quartile Range is called Quartile Deviation (Q.D.). Thus:

Q. D =Q₃ - Q₁ /2

Q.D. is therefore also called Semi- Inter Quartile Range.

Example 1

Calculate Range and Q.D. of the following observations:

20, 25, 29, 30, 35, 39, 41, 48, 51, 60 and 70

Range is clearly 70 – 20 = 50

For Q.D., we need to calculate values of Q₃ and Q₁ .

Q₁ is the size of n+1/4  th value.

n being 11, Q₁ is the size of 3^rd value.

As the values are already arranged in ascending order, it can be seen that Q₁, the 3rd value is 29. [What will you do if these values are not in an order?]

Similarly, Q₃ is size of  3(n+1)/4 th value; i.e. 9^th value which is 51. Hence, Q₃=51.

Q. D. =Q₃ - Q₁ /2 = 51-29/2  = 11.

(c) Mean Deviation

Suppose a college is proposed for students of five towns A, B, C, D and E which lie in that order along a road. Distances of towns in kilometers from town A and number of students in these towns are given below:

Now, if the college is situated in town A, 150 students from town B will have to travel 2 kilometers each (a total of 300 kilometers) to reach the college. The objective is to find a location so that t h e   average distance travelled by students is minimum. You may observe that the students will have to travel more, on an average, if the college is situated at town A or E. If on the other hand, it is somewhere in the middle, they are likely to travel less. Mean deviation is the appropriate statistical tool to estimate the average distance travelled by students. Mean Deviation is the arithmetic mean of the differences of the values from their average.  The average used is either the arithmetic mean or median.

(Since the mode is not a stable average, it is not used to calculate Mean Deviation.)

A proper approach to the measurement of dispersion or variability would require that all the values in a series are taken into consideration. One of the methods of doing it is through average deviation or mean deviation.

Following mathematical formula is used to estimate mean deviation of a data series of mean deviation of a data series,

Case I: Ungrouped data series

Example

Compute mean deviation from the following data

Solution           Computation of Mean Deviation