Measures of Dispersion | Meaning, Absolute and Relative Measures of Dispersion

Last Updated : 25 Jul, 2024

Averages like mean, median, and mode can be used to represent any series by a single number. However, averages are not enough to describe the characteristics of statistical data. So, it is necessary to define some additional summary measures to adequately represent the characteristics of a distribution. One such measure is ' Measures of Dispersion.' Measures of Dispersion is the calculation of the extent to which numerical data is likely to vary about an average value.

Measures of Dispersion

Dispersion is the extent to which values in a distribution differ from the average of the distribution. Dispersion indicates a lack of uniformity in the size of items. It is the calculation of the extent to which numerical data is likely to vary about an average value. There are certain specific measures that help in determining the deviations from the central value, including Range, Quartile Deviations, Interquartile Deviations and Semi-Interquartile Deviations, Mean Deviations, Lorenz Curve, Standard Deviation, etc.

Dispersion or spread is the degree of the scatter or variation of the variables about a central value.
-Brooks and Dick

Objectives of Dispersion

Measures of Dispersions are used in computations of various important statistical techniques like correlation, regression, hypothesis test, etc.
The study of variation helps to analyse the reasons and causes of variations. This may be helpful in controlling the variation itself.
It aims to find out the degree of uniformity/consistency in two or more sets of data. A high degree of variation would mean little uniformity/consistency, whereas a low degree of variation would mean greater uniformity/consistency.
Measures of dispersion are used to test to what extent, an average represents the characteristic of a series.

Dispersion: Absolute or Relative Measures

There are two main types of measures of Dispersion - Absolute Dispersion and Relative Dispersion

Absolute Measures: The measures of dispersion which are expressed in terms of original units of a series are termed as Absolute Measures. Absolute Measures are expressed in concrete units; i.e., units in terms of which the data has been expressed like rupees, centimeters, kilograms, etc. Such measures are not suitable for comparing the variability of the two distributions which are expressed in different units of measurement.
Relative Measures: The measures of dispersion which are measured as a percentage or ratio of the average are termed Relative Measures. Relative Measures are sometimes known as coefficients of dispersion as 'coefficient' means a pure number that is independent of a unit of measurement.

There are certain specific measures that help in determining the deviations from the central value, including Range, Quartile Deviations, Interquartile Deviations and Semi-Interquartile Deviations, Mean Deviations, Lorenz Curve, Standard Deviations, etc.

1. Range:

Range is the simplest of all the measures of dispersion. Range (absolute measure) is defined as the numerical difference between the largest and the smallest item of a distribution. It cannot be usually employed to compare the variability of two distributions expressed in different units, like the amount of dispersion measured in rupees is not comparable with dispersion measured in inches.

Range (R) = Largest item (L) - Smallest item (S)

Coefficient of Range (relative measure) refers to the ratio of the difference between two extreme items, the largest and the smallest, of the distribution to their sum.

Coefficient of Range = \frac{Largest~item~(L)-Smallest~item(S)}{Largest~item~(L)+Smallest~item(S)}

2. Quartile Deviation:

Quartiles are the values that create quarters from the complete list. Quartile Deviation (absolute measure) divides the distribution into multiple quarters. Quartile Deviation is further divided into two categories: Interquartile Range and Semi-Interquartile Range. Interquartile Range refers to the difference between the values of two quartiles, and the Semi-Interquartile Range is expressed as half of the difference between the upper quartile and the lower quartile.

Interquartile Range = Q₃- Q₁

Semi-Interquartile Range = \frac{Q_3-Q_1}{2}

For comparative studies of the variability of two distributions, the Coefficient of Quartile Deviation (relative measure) is used.

Coefficient of Quartile Deviation = \frac{Q_3-Q_1}{Q_3+Q_1}

3. Mean Deviation/First Moment of Dispersion/Average Deviation:

Mean Deviation (absolute measure) of a series is the arithmetic average of the deviations of various items from a measure of central tendency (mean, median, or mode). Mean Deviation is based on all the items of the series. In order to convert it to a relative measure; i.e., Coefficient of Mean Deviation, it is divided by the average, from which it has been calculated.

Mean (\bar{X} ) = \frac{\sum{X}}{N}

Mean Deviation from Mean (MD_{\bar{X}}) = \frac{\sum{|D|}}{N}

Coefficient of Mean Deviation = \frac{MD_{\bar{X}}}{\bar{X}}

4. Standard Deviation:

Standard Deviation (absolute measure) is the square root of the arithmetic average of the squares of the deviations measured from the mean. Standard Deviation is also known as root mean square deviation because it is the square root of the means of squared deviations from the arithmetic mean. Coefficient of Standard Deviation (relative measure) is used for the comparison of dispersions of two or more series with different units and is calculated by dividing the standard deviation by the mean of the data.

5. Lorenz Curve:

Lorenz Curve (absolute measure) is a graphic method of studying dispersion. This curve is usually used to measure the inequalities of income or wealth in a society. If the distribution is uniform, the Lorenz curve will coincide with the line of equal distribution. The extent of deviation of the actual distribution of a statistical series from the line of equal distribution is known as the Lorenz Coefficient (relative measure).

Range | Meaning, Coefficient of Range, Merits and Demerits, Calculation of Range

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Measures of Dispersion | Meaning, Absolute and Relative Measures of Dispersion