Prof. Rajkumar TeotiaInstitute of Advanced Management and Research (IAMR)

Address: 9th Km Stone, NH-58, Delhi-Meerut Road, Duhai,Ghaziabad (U.P) - 201206

Ph:0120-2675904/905 Mob:9999052997 Fax: 0120-2679145e mail: rajkumarteotia@iamrindia.com

MEASURES OF DISPERSION OR VARIATION

Dispersion measures the extent to which the items vary from some central value. It may be noted that the measures of dispersion measure only the degree (the amount of variation) but not the direction of variation.

  The various measures of central value give us one single

figure that represents the entire data. But the average alone cannot adequately describe a set of observations, unless all the observations are the same. It is necessary to describe the variability or dispersion of the observations.

MEASURES OF DISPERSION OR VARIATION

A good measure of dispersion should possess the following properties

It should be simple to understand. It should be easy to compute. It should be rigidly defined. It should be based on each and every item of the distribution. It should be amenable to further algebraic treatment. It should have sampling stability. Extreme items should not unduly affect it.

SIGNIFICANCE AND PROPERTIES OF MEASURES OF DISPERSION OR VARIATION

Range Inter-quartile range or Quartile Deviation Mean deviation or Average Deviation Standard Deviation Lorenz curve.

There are five measures of dispersion:

Range is defined as the difference between the value of largest item and the value of smallest item included in the distribution. Measures of range may be absolute or relative. Formula for calculating range is

Range = L – S Where, L = Largest value S = Smallest Value

Coefficient of range = L – S L + S

1. Range

MERITS OF RANGE

It should be simple to understand. It should be easy to compute. It should be rigidly defined

LIMITATIONS OF RANGE:-  It is based only on two items and does not cover all the items in a

distribution. It is subject to wide fluctuations from sample to sample based on the

same population. It fails to give any idea about the pattern of distribution. Finally, in the case of open-ended distributions, it is not possible to

compute the range.  

Quartile deviation is half of the difference between upper

quartile (Q3) and lower quartile (Q1). Quartile deviation indicates the average amount by which the two quartiles differ from the median. In symmetrical distribution the two quartiles (Q1 and Q3) are equidistant from the median.

Symbolically, inter quartile range = Q3- Ql

2.Inter-quartile range or Quartile Deviation

Many times the inter quartile range is reduced in the form of semi-inter quartile range or quartile deviation as shown below:

Semi inter quartile range or Quartile deviation = (Q3 – Ql) 2

Coefficient of Quartile Deviation = (Q3 – Ql) (Q3 + Ql )

 

MERITS OF QUARTILE DEVIATION The following merits are entertained by quartile deviation:

As compared to range, it is considered a superior measure of dispersion.

In the case of open-ended distribution, it is quite suitable.

Since it is not influenced by the extreme values in a distribution, it is particularly suitable in highly skewed or erratic distributions.

 .

LIMITATIONS OF QUARTILE DEVIATION  Like the range, it fails to cover all the items in a distribution.

It is not amenable to mathematical manipulation.

It varies widely from sample to sample based on the same population.

Since it is a positional average, it is not considered as a measure of dispersion. It merely shows a distance on scale and not a scatter around an average. In view of the above-mentioned limitations, the inter quartile range or the quartile deviation has a limited practical utility

Average deviation is obtained by calculating the absolute deviations of each observation from median or mean and then averaging these deviations by taking their arithmetic mean. Average deviation is denoted by A.D

 

3.Mean deviation or Average Deviation:-

In case deviation is taken from median:- 

A.D Median

Computation of average deviation for ungrouped data:-

A.D (Median) = ∑ X - Median  N

Coefficient of A.D (Median) = 

In case deviation is taken from mean:-

A.D (Mean) = ∑ X - Mean  N

Coefficient of A.D (Mean) = A.D Mean 

In case deviation is taken from median:-

Computation of average deviation for grouped data:-

A.D (Median) = ∑f X - Median  N

Coefficient of A.D (Median) = A.D Median 

In case deviation is taken from mean:-

A.D (Mean) = ∑f X - Mean  N

Coefficient of A.D (Mean) = A.D Mean 

MERITS OF MEAN DEVIATION  A major advantage of mean deviation is that it is simple to

understand and easy to calculate.

It takes into consideration each and every item in the distribution. As a result, a change in the value of any item will have its effect on the magnitude of mean deviation.

The values of extreme items have less effect on the value of the mean deviation.

As deviations are taken from a central value, it is possible to have meaningful comparisons of the formation of different distributions.

LIMITATIONS OF MEAN DEVIATION  It is not capable of further algebraic treatment.

At times it may fail to give accurate results. The mean deviation gives best results when deviations are taken from the median instead of from the mean. But in a series, which has wide variations in the items, median is not a satisfactory measure.

Strictly on mathematical considerations, the method is wrong as it ignores the algebraic signs when the deviations are taken from the mean.

The standard deviation measures the absolute variation of a distribution. The greater the amount of variation, the greater the standard deviation, for the greater will be the magnitude of the deviation on the values from their mean.

A small standard deviation means a high degree of uniformity of the observations as well as homogeneity of a series, a large standard deviation means just opposite.

Hence standard deviation is extremely useful in judging the representativeness of the mean. It is denoted by the small Greek letter σ (read as sigma).

4. STANDARD DEVIATION

In case deviation is taken from Actual mean:-

Computation of Standard deviation for Ungrouped data:-

σ = ∑ (X – X) 2 N

Where d = X - A

In case deviation is taken from Assumed mean:-

σ = ∑ d 2 – ∑ d 2 N N

Computation of Standard deviation for grouped data:- In case deviation is taken from Actual mean:-

σ = ∑ f (X – X) 2 N

Where d = X – A i

In case deviation is taken from Assumed mean:-

σ = ∑ fd 2 – ∑ fd 2 x i N N

Coefficient of variation:-

 Coefficient of variation is calculated by the following formula

Where, C.V = coefficient of variation σ = standard deviation. X = Arithmetic Mean. 

  

C.V = σ x 100

X

Relationship between standard deviation and variance If we square standard deviation we get variance

OR

σ = Variance

Variance = σ2

Just as it is possible to compute combined mean of two or more than two groups. Similarly we can also compute combined standard deviation of two groups is denoted by the following formula.

σ12 = N1σ12 + N2σ2

2 + N1d12 + N2d2

2

N1 + N2

Combined Standard deviation:-

Where,  σ12 = combined standard deviation of two groups.

σ1 = standard deviation of first group.

σ2 = standard deviation of second group

N1 = number of items in first group. N2 = number of items in second group.

 

d1 = X1 – X12 And d2 = X2 – X12

 

PROPERTIES OF THE STANDARD DEVIATION:- The sum of squares of the deviation of all observation from arithmetic mean is

Minimum.  The standard deviation of the first n natural numbers can be obtained by the following

formula: 

σ = 1 (N 2 – 1) 12

Thus standard deviation of natural numbers 1 to 10 will be 

σ = 1 (10 2 – 1) = 2.87 12  

Standard deviation is independent of change of origin but not scale.  For a symmetrical distribution, the following area relationships

hold good.  It enables us to determine as to how far individual items in a distribution

deviate from its mean. In a symmetrical, bell-shaped curve:

(i) About 68.27 percent of the values in the population fall within: + 1 standard deviation from the mean.

 (ii) About 95.45 percent of the values will fall within +2 standard

deviations from the mean. (iii) About 99.73 percent of the values will fall within + 3 standard

deviations from the mean. 

68.27%

95.45%

X - 3σ X - 2σ X - σ X X + σ X + 2σ X - 3σ

This can be illustrated by the following diagram

This measure of dispersion is graphical. It is known as the Lorenz curve named after Dr. Max Lorenz. It is generally used to show the extent of concentration of income and wealth. The steps involved in plotting the Lorenz curve are:

  Convert a frequency distribution into a cumulative frequency

table.  On the X – axis, start from 0 to 100 and take the percent of

variable.  On the Y – axis, start from 0 to 100 and take the percent of

variable. 

5.LORENZ CURVE

Draw a diagonal line joining 0 with 100. This is known as line of equal distribution. Any point on this line shows the same percent on X as on Y

  Plot the various points corresponding to X and Y and join

them. The distribution so obtained, unless it is exactly equal, will always curve below the diagonal line. If two curves of distribution are shown on the same Lorenz presentation, the curve that is farthest from the diagonal line represents the greater inequality.

Figure 3.1 shows two Lorenz curves by way of illustration. The straight line AB is a line of equal distribution, whereas AEB shows complete inequality. Curve ACB and curve ADB are the Lorenz curves