Dispersion Measures and Lorenz Curve Overview

Relative Measure of Dispersion is one of the most important chapters in Statistics or Mathematical Economics. Various distributions are compared with the help of absolute and relative measures of dispersion. In the following article, we aim at discussing an absolute and relative measure of dispersion along with the Lorenz curve, a graphical measure of dispersion. Absolute and relative dispersion have numerous uses in the field of income distribution, wealth distribution, profits and wages distribution etc.

Relative Measures of Dispersion

A given series of data is accurately exhibited by the absolute measures of dispersion. But one of the major demerits of this is that if there is a need to compare dispersion for a series of different units then it cannot be used. The above-mentioned comparison can be done with relative dispersion.

Dispersion is of two types- absolute and relative dispersion. Absolute dispersion and relative dispersion are the tools to perform complete enumeration and relative comparison respectively. Absolute and relative measures of dispersion are correlated to each other. According to the relative dispersion definition, the dispersed data are expressed in some relative terms or percentage. It is possible to compare the various series as they are devoid of a particular unit. Relative Measure of Dispersion is a subset in the absolute measure of dispersion.

The Relative Measure of Dispersion formula can be derived by the ratio of absolute variability to the mean value or by the percentage of absolute variability. Another name of relative measures of dispersion is coefficients of dispersion. The following relative measures of variation will be briefly discussed:

• Coefficient of range.

• Coefficient of quartile deviation.

• The coefficient of mean deviation.

• Coefficient of standard deviation and coefficient of variation.

Relative Measure of Dispersion Formula

Coefficient of Range:

(H - L)/(H + L)

H = The highest value

L = The lowest value

Coefficient of Quartile Deviation:

(Q3 - Q1)/(Q3 + Q1)

Q3 = Third quartile

Q1 = First quartile

First and third quartile for individual series is calculated by the formula:

Q1= Size of (N + 1)/4th item and Q3 = Size of 3(N + 1)/4th item. Here N stands for the number of observations.

First and third quartile of discrete series is calculated as follows:

Primarily a column of cumulative frequency is formed based on each observation. Then the values of (N+1)/4 and 3(N+1)/4 are calculated based on the calculation of Q1 and Q3. Here, N stands for the summation of frequencies.

Coefficient of Mean Deviation:

Coefficient of Mean Deviation About Mean: (mean deviation about mean)/arithmetic mean.

Coefficient of Mean Deviation About Median: (mean deviation about median)/ median.

Coefficient of Mean Deviation About Mode: (mean deviation about mode)/ mode.

Coefficient of Standard Deviation:

Coefficient of Standard Deviation: σ/Mean

Here, σ= Standard deviation for the series.

Coefficient of Variation:

Coefficient of Variation: (Coefficient of standard deviation) X 100

Lorenz Curve:

The absolute measure of dispersion is measured graphically by the Lorenz curve. The actual curve and a line of equal distribution are represented graphically through the Lorenz curve. It displays the deviation between these two.

The divergence of an actual curve from the line of equal distribution is called Lorenz Coefficient. It is positively correlated with the distance of the Lorenz curve from the line of equal distribution.

(image will be uploaded soon)

Did You Know?

• In the frequency distribution series, Q2 or the second quartile is also known as the median.

• Median is calculated in the same way as Q1 and Q3. Only, there is no usage of the term N/2.

• Construction of Lorenz curve is dependent upon two factors namely cumulative percentage for observation and cumulative percentage for frequency.

• While constructing Lorenz curve cumulative frequencies are plotted in the X-axis and cumulative items are plotted in the Y-axis.

• In the Lorenz, curve values commence from 0 to 100.

• The curve is a straight line with an inclination of 45 degrees to both the axes and connecting the origin to the point (100, 100).

• A good measure of dispersion is very effortless to calculate and easy to understand.

• Sampling fluctuations cannot always affect a good measure of dispersion.

The absolute measures of dispersion are as follows:

1. Range

2. Interquartile Range

3. Quartile Deviation

4. Mean Deviation

5. Standard Deviation

6. Lorenz curve 

From the above article, various absolute and relative measures of dispersion are vividly discussed. Absolute and relative dispersion is used in calculating several factors.