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Measures of Dispersion in Statistics | Complete UG Notes with Examples

Measures of Dispersion – Complete Statistics Notes

In Statistics, understanding data does not end with finding an average. Two different data sets may have the same average but still behave very differently. To study how data values are spread or scattered around an average, we use Measures of Dispersion.

Measures of dispersion tell us about the variability, consistency, and reliability of data. These measures are extremely important in real-life decision making, economics, quality control, education, and scientific research.

1. Meaning of Dispersion

The word dispersion means spread or scattering. In statistics, dispersion refers to the extent to which observations in a data set differ from one another and from the central value (mean, median, or mode).

If data values are very close to each other, dispersion is said to be low. If data values are widely spread, dispersion is high.

Example:

Data A: 10, 10, 10, 10, 10
Data B: 2, 6, 10, 14, 18

Both data sets have the same mean (10), but Data A has no variation while Data B has large variation. This difference is explained using measures of dispersion.

2. Objectives of Measuring Dispersion

Measures of dispersion are used for the following purposes:

  • To understand the spread of data
  • To measure consistency or variability
  • To compare two or more data series
  • To test the reliability of averages
  • To help in decision making in economics and business

3. Types of Measures of Dispersion

Measures of dispersion are broadly classified into two types:

  • Absolute Measures of Dispersion
  • Relative Measures of Dispersion

4. Absolute Measures of Dispersion

Absolute measures express dispersion in the same units as the data. The main absolute measures are:

  • Range
  • Quartile Deviation
  • Mean Deviation
  • Standard Deviation

4.1 Range

Range is the simplest measure of dispersion. It is defined as the difference between the highest and the lowest values in a data set.

Formula:

Range = Maximum Value − Minimum Value

Example:

Data: 5, 9, 12, 15, 20
Range = 20 − 5 = 15

Merits of Range

  • Easy to calculate
  • Quick measure of spread

Demerits of Range

  • Based only on two values
  • Highly affected by extreme values
  • Not reliable for detailed analysis

4.2 Quartile Deviation (Semi-Interquartile Range)

Quartile Deviation measures dispersion based on the middle 50% of data. It is less affected by extreme values.

Formula:

Quartile Deviation (Q.D.) = (Q3 − Q1) / 2

Where:
Q1 = First Quartile
Q3 = Third Quartile

Merits

  • Less affected by extreme values
  • Useful for skewed distributions

Demerits

  • Ignores half of the data
  • Not suitable for algebraic treatment

4.3 Mean Deviation

Mean Deviation is the average of absolute deviations of observations from a central value (mean, median, or mode).

Formula (about Mean):

Mean Deviation = Σ|x − x̄| / n

Mean deviation is always taken as positive, ignoring signs.

Merits

  • Based on all observations
  • Simple to understand

Demerits

  • Absolute values make calculation difficult
  • Limited use in theoretical statistics

4.4 Standard Deviation

Standard Deviation is the most important and widely used measure of dispersion. It measures the square root of the average of squared deviations from the mean.

Formula:

σ = √[ Σ(x − x̄)² / n ]

A smaller standard deviation indicates greater consistency, while a larger standard deviation shows more variability.

Merits

  • Based on all observations
  • Mathematically precise
  • Widely used in advanced statistics

Demerits

  • Calculation is lengthy
  • Affected by extreme values

5. Relative Measures of Dispersion

Relative measures are used to compare variability between different data sets. They are expressed in ratios or percentages.

  • Coefficient of Range
  • Coefficient of Quartile Deviation
  • Coefficient of Mean Deviation
  • Coefficient of Variation (C.V.)

Coefficient of Variation:

C.V. = (σ / x̄) × 100

A lower C.V. indicates greater consistency.

6. Importance of Measures of Dispersion

  • Helps compare two distributions
  • Measures data reliability
  • Used in quality control
  • Important in economics and business decisions

7. Practice Questions

  1. Define measures of dispersion.
  2. Why are measures of dispersion important?
  3. Calculate range for: 12, 15, 20, 30, 35.
  4. State merits and demerits of standard deviation.
  5. Differentiate between absolute and relative measures.

8. Conclusion

Measures of dispersion play a vital role in statistical analysis by showing how data values are spread around a central value. They complement measures of central tendency and help in understanding the true nature of data.

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