Relative Measures of Dispersion: Threshold of Core Statistics
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:
Range
Interquartile Range
Quartile Deviation
Mean Deviation
Standard Deviation
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.
FAQs on Dispersion Measures and Lorenz Curve Overview
1. What are measures of dispersion in statistics?
Measures of dispersion are statistical tools used to quantify the extent to which data points in a series are spread out or scattered around an average value, such as the mean or median. They provide a measure of the variability or consistency within a dataset. A low dispersion indicates that the data points tend to be close to the average, while a high dispersion signifies that they are spread over a wider range.
2. What is the main difference between absolute and relative measures of dispersion?
The main difference lies in their units and purpose. Absolute measures express variability in the same units as the original data (e.g., rupees, kilograms), making them useful for understanding dispersion within a single dataset. Relative measures, on the other hand, are unit-free ratios or percentages, making them ideal for comparing the variability of two or more datasets that may have different units or different average values.
3. What are the most common absolute measures of dispersion as per the CBSE syllabus?
According to the CBSE Class 11 Economics syllabus for 2025-26, the key absolute measures of dispersion that students need to understand are:
- Range: The difference between the highest and lowest values.
- Quartile Deviation: Half of the difference between the upper quartile (Q3) and the lower quartile (Q1).
- Mean Deviation: The arithmetic average of the absolute deviations of items from a measure of central tendency (mean, median, or mode).
- Standard Deviation: The square root of the arithmetic mean of the squares of the deviations of items from their mean value.
4. What is a Lorenz Curve and what does the ‘line of equal distribution’ represent?
A Lorenz Curve is a graphical representation of the distribution of a variable, commonly used in economics to illustrate income or wealth inequality. The ‘line of equal distribution’ is a straight diagonal line at a 45-degree angle. It represents a scenario of perfect equality, where each cumulative percentage of the population holds the same cumulative percentage of the total income or wealth (e.g., the bottom 20% of households holds 20% of the income).
5. How does the Lorenz Curve help in visually analysing income or wealth inequality?
The Lorenz Curve makes inequality easy to see. The degree of inequality is represented by the gap between the line of equal distribution and the actual Lorenz Curve. A Lorenz Curve that is very close to the diagonal line indicates a low level of inequality. Conversely, a curve that bows far away from the line signifies a high degree of inequality, showing that a small percentage of the population holds a large percentage of the total wealth or income.
6. Why can't an absolute measure like 'range' be used to compare the variability of two different datasets, such as the heights of students and their exam scores?
Absolute measures like range cannot be used for such a comparison because they are expressed in the original units of the data. The range for student heights would be in centimetres, while the range for exam scores would be in marks. Comparing 'centimetres' to 'marks' is meaningless. This is why relative measures of dispersion, like the Coefficient of Variation, are necessary, as they are unit-free and allow for a standardised comparison of variability across different types of data.
7. Why is the Coefficient of Variation often considered the most reliable tool for comparing the consistency between different data series?
The Coefficient of Variation (CV) is considered highly reliable for comparison because it is a relative measure that standardises the standard deviation by expressing it as a percentage of the mean. This process removes the influence of the scale of measurement and the average value of the series. Therefore, it allows for a fair comparison of consistency, even between datasets with vastly different means (e.g., comparing the price fluctuation of gold with that of onions).
8. What are the key characteristics that a good measure of dispersion should possess?
A good measure of dispersion should ideally have the following characteristics:
- It should be easy to understand and simple to calculate.
- It must be rigidly defined with a clear mathematical formula.
- It should be based on all observations in the dataset.
- It must be capable of further algebraic treatment for more advanced statistical analysis.
- It should be least affected by sampling fluctuations and not be unduly influenced by extreme values.
