Introduction to Measures of Dispersion

The Limitations of Averages

In the previous chapter, you learned how to summarize a dataset into a single representative value using measures of central tendency (like mean, median, and mode). However, this single value does not reveal the variability or dispersion present in the data. An average tells us about the central location of a distribution but says nothing about how the values are spread out around that central point.

This example clearly shows that knowledge of only the average is insufficient. We need another value that reflects the quantum of variation to better understand the distribution.

What is Dispersion?

Dispersion is the extent to which values in a distribution differ from the average of that distribution. It measures the spread or variability of the data. A measure of dispersion can reveal information about inequalities. For example, while per capita income gives an average, a measure of dispersion can tell us about income inequalities between different strata of society.

Common Measures of Dispersion

To quantify the extent of variation, there are several measures:

1. Range
2. Quartile Deviation
3. Mean Deviation
4. Standard Deviation

Range and Quartile Deviation are measures based on the spread of values, while Mean Deviation and Standard Deviation are based on the deviations of values from an average.

Measures Based on the Spread of Values

These measures calculate dispersion by looking at the spread within which the values of a distribution lie.

1. Range

The Range (R) is the simplest measure of dispersion. It is the difference between the largest (L) and the smallest (S) value in a distribution.

$R = L - S$

A higher value of the range implies higher dispersion.

Limitations of Range

• It is unduly affected by extreme values (outliers).
• It is not based on all the values in the distribution; it only uses the two extreme observations.
• It cannot be calculated for open-ended frequency distributions (where the lowest or highest class limit is not specified).

Despite its limitations, the range is frequently used because of its simplicity. For example, we often see the maximum and minimum temperatures of cities, which is a form of range.

2. Quartile Deviation

To overcome the problem of extreme values, the Quartile Deviation (Q.D.) is used. It is based on the middle 50% of the data and is therefore not affected by outliers. It is calculated using the upper quartile ($Q_3$) and the lower quartile ($Q_1$).

First, the Interquartile Range is calculated, which is the difference between the third and first quartiles:

Interquartile Range = $Q_3 - Q_1$

Half of the interquartile range is called the Quartile Deviation (Q.D.), also known as the Semi-Interquartile Range.

Quartile Deviation (Q.D.) = $\frac{Q_3 - Q_1}{2}$

Calculation of Quartile Deviation

The process involves finding the values of $Q_1$ and $Q_3$.

• For ungrouped data:

  $Q_1 =$ size of $(\frac{n+1}{4})^{th}$ item

  $Q_3 =$ size of $3(\frac{n+1}{4})^{th}$ item

• For continuous series:

  First, find the quartile class where the $(\frac{n}{4})^{th}$ item (for $Q_1$) or $(\frac{3n}{4})^{th}$ item (for $Q_3$) lies. Then, use the formula:

  $Q_1 = L + \frac{(\frac{n}{4} - c.f.)}{f} \times i$

  Where L is the lower limit of the quartile class, c.f. is the cumulative frequency of the preceding class, f is the frequency of the quartile class, and i is the class interval.

Quartile deviation can be calculated for open-ended distributions and is a good measure of dispersion when extreme values are present.

Measures of Dispersion from Average

Range and Quartile Deviation give an idea about the spread of values but do not measure how far the values are from their average. Two important measures based on the deviation of values from their average are Mean Deviation and Standard Deviation.

Since the average is a central value, some deviations from it are positive and some are negative. The sum of deviations from the Arithmetic Mean is always zero. To overcome this, Mean Deviation ignores the signs of the deviations, while Standard Deviation squares them.

1. Mean Deviation (M.D.)

The Mean Deviation is the arithmetic mean of the absolute differences (deviations) of the values from their average. The average used can be either the arithmetic mean or the median.

Calculation of Mean Deviation

(a) From Arithmetic Mean for ungrouped data:

M.D. $(\bar{x}) = \frac{\sum|d|}{n} = \frac{\sum|X - \bar{X}|}{n}$

(b) From Median for ungrouped data:

M.D. (Median) = $\frac{\sum|d|}{n} = \frac{\sum|X - \text{Median}|}{n}$

(c) For continuous distribution (from mean):

M.D. $(\bar{x}) = \frac{\sum f|d|}{\sum f} = \frac{\sum f|m - \bar{X}|}{\sum f}$

Limitations of Mean Deviation

Mean deviation is based on all values. However, its main limitation is that it ignores the algebraic signs of the deviations, which makes it appear unmathematical and less suitable for further statistical analysis.

2. Standard Deviation (S.D.)

The Standard Deviation is the most widely used measure of dispersion. It is defined as the positive square root of the mean of the squared deviations from the arithmetic mean. It is denoted by the Greek letter sigma ($\sigma$).

Variance and Standard Deviation

The mean of the squared deviations from the mean is called the Variance ($\sigma^2$). The Standard Deviation is the positive square root of the variance.

Variance ($\sigma^2$) = $\frac{\sum(X - \bar{X})^2}{n}$

Standard Deviation ($\sigma$) = $\sqrt{\frac{\sum(X - \bar{X})^2}{n}}$

Calculation of Standard Deviation (for continuous frequency distribution)

Several methods can be used, with the step-deviation method being one of the most common for simplifying calculations.

$\sigma = \sqrt{\frac{\sum fd'^2}{\sum f} - \left(\frac{\sum fd'}{\sum f}\right)^2} \times c$

Where $d' = \frac{m - A}{c}$ (m = midpoint, A = assumed mean, c = common factor).

Properties of Standard Deviation

• It is based on all values.
• It is always calculated from the mean.
• It is independent of origin (a change in the origin does not affect S.D.) but not of scale (if values are divided/multiplied by a constant, the S.D. is also divided/multiplied by that constant).
• It is suitable for further algebraic treatment and is used in many advanced statistical problems.

Absolute and Relative Measures of Dispersion

Absolute vs. Relative Measures

All the measures discussed so far (Range, Q.D., M.D., S.D.) are absolute measures of dispersion. They express the variation in the same units as the original data (e.g., rupees, kilograms).

Weaknesses of Absolute Measures

• Difficult to Compare: They can be misleading when comparing the variability of two different distributions, especially when their averages differ significantly or their units of measurement are different. For instance, a range of ₹500 in the sales of a small vendor is not comparable to a range of ₹30,000 for a large departmental store.
• Unit Dependent: The value of the measure changes if the unit of measurement changes (e.g., from kilometers to meters).

Relative Measures of Dispersion

To overcome these problems, relative measures of dispersion are used. These are expressed as ratios or percentages and are free from the units of measurement, making them suitable for comparison. Each absolute measure has a corresponding relative measure, often called a "coefficient".

The Coefficient of Variation (C.V.) is the most commonly used relative measure of dispersion. It is expressed as a percentage and is used to compare the variability, consistency, or uniformity of two or more series. A series with a lower C.V. is considered more consistent or stable.

Lorenz Curve: A Graphic Measure of Dispersion

The Lorenz Curve is a graphical method used to estimate and visualize inequalities in a distribution, particularly for income and wealth. While other measures provide a numerical value of dispersion, the Lorenz curve provides a visual representation.

Construction of the Lorenz Curve

The following steps are required to construct a Lorenz Curve:

1. Data on the variable (e.g., income) and the number of individuals (e.g., employees) are arranged in classes.
2. The cumulative frequencies of the number of individuals and the cumulative values of the variable are calculated.
3. These cumulative values are then converted into percentages of their respective totals.
4. The cumulative percentages of the individuals are plotted on the horizontal axis (X-axis), and the cumulative percentages of the variable are plotted on the vertical axis (Y-axis).
5. The plotted points are joined by a smooth curve. This curve is the Lorenz Curve.
6. A straight diagonal line is drawn from the origin (0, 0) to the point (100, 100). This is called the Line of Equal Distribution.

Studying the Lorenz Curve

The Line of Equal Distribution represents a situation of perfect equality (e.g., the bottom 20% of people earn 20% of the total income, the bottom 50% earn 50%, and so on). The farther the Lorenz Curve is from this line, the greater the inequality present in the distribution.

The Lorenz Curve is especially useful for comparing the degree of inequality in two or more distributions by drawing their curves on the same graph. The curve that is farthest from the line of equal distribution represents the distribution with the highest inequality.

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