Kerala Plus One Economics Notes Chapter 16 Measures of Dispersion
Dispersion
A measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative standards of living enjoyed by different strata of society. Dispersion is the extent to which values in a distribution differ from the average of the distribution.
To quantify the extent of the variation, there are certain measures namely:
- Range
- Quartile Deviation
- Mean Deviation
- Standard Deviation
- Lorenz Curve
Range
Range (R) is the difference between the largest (L) and the smallest value (S) in a distribution.
Thus, R = L – S
A higher value of Range implies higher dispersion and vice-versa.
Quartile Deviation
The presence of even one extremely high or low value in distribution can reduce the utility of range as a measure of dispersion. Thus, you may need a measure that 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.
Mean Deviation
The mean deviation of a series is the arithmetic average of the deviations of various items from a measure of central tendency. In aggregating the deviations, algebraic signs of the deviations are not taken into account. It is because, if the algebraic signs were taken into account, the sum of deviations from the mean should be zero and that from the median is nearly zero. Theoretically, the deviations can be taken from any of the three averages, namely, arithmetic mean, median, or mode; but, the mode is usually not considered as it is less stable. Between mean and median, the latter is supposed to be better because the sum of the deviations from the median is less than the sum of the deviations from the mean,
Co-efficient of MD = \(\frac{\mathrm{MD}}{\text { Average }}\)
Standard Deviation
Standard deviation is defined as the square root of the arithmetic average of the squares of deviations taken from the arithmetic average of a series. It is also known as the root-mean-square deviation for the reason that it is the square root of the mean of the squared deviations from AM.
Standard deviation is denoted by the Greek letter a (small letter ‘sigma’). The term variance is used to describe the square of the standard deviation. Standard deviation is an absolute measure of dispersion. The corresponding relative measure is called the coefficient of SD. The coefficient of variation is also a relative measure. A series with more coefficient of variation is regarded as less consistent or less stable than a series with less coefficient of variation.
Symbolically, Standard deviation = σ
Variance = σ2
Coefficient of SD = \(\frac{\sigma}{\bar{x}}\)
Coefficient of variation = \(\frac{\sigma}{\bar{x}} \times 100\)
Lorenz Curve
The measures of dispersion discussed so far give a numerical value of dispersion. A graphical measure called the Lorenz Curve is available for estimating dispersion.
Lorenz Curve uses the information expressed in a cumulative manner to indicate the degree of variability. It is especially useful in comparing the variability of two or more distributions.
Construction of the Lorenz Curve
Following steps are required for the Construction of the Lorenz Curve
- Calculate class mid-points and find cumulative totals
- Calculate cumulative frequencies
- Express the grand totals and convert the cumulative totals into percentages,
- Now, on the graph paper, take the cumulative percentages of the variable (incomes) on Y-axis and cumulative percentages of frequencies (number of employees) on the X-axis.
- Draw a line joining Co-ordinate (0, 0) with (100, 100). This is called the line of equal distribution.
- Plot the cumulative percentages of the variable with corresponding cumulative percentages of frequency. Join these points to get the curve.