Measures of Dispersion

Dispersion (or variability) tells us how spread out or scattered the data is around a central value (like mean or median). While central tendency measures the “center,” dispersion measures the “spread.”

Dispersion is important because two datasets can have the same mean but different spreads.

1. Absolute Measures of Dispersion

Absolute measures express dispersion in the same units as the data. They show how much the data deviates from a central value in absolute terms.

Common Absolute Measures

1. Range $$\text{Range} = \text{Maximum value} - \text{Minimum value}$$

• Simple but ignores intermediate values.

2. Quartile Deviation (QD) / Semi-Interquartile Range $$QD = \frac{Q_3 - Q_1}{2}$$

• Based on middle 50% of data.
• Less affected by extreme values than range.

3. Mean Deviation (MD) / Average Deviation $$MD = \frac{\sum |x - \bar{x}|}{N}$$

• Average of absolute differences from mean (or median).

4. Variance $$\sigma^2 = \frac{\sum (x - \bar{x})^2}{N} \quad \text{(Population)}$$

• Square of deviations from mean.

5. Standard Deviation (SD) $$\sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{N}}$$

• Square root of variance.
• Most widely used measure of dispersion.

2. Relative Measures of Dispersion

Relative measures express dispersion in relation to the central value, usually as a percentage. They are unit-free, making it easier to compare variability between datasets.

Common Relative Measures

1. Coefficient of Variation (CV) $$CV = \frac{\text{SD}}{\text{Mean}} \times 100$$

• Measures variability relative to the mean.
• Useful for comparing datasets with different units or scales.

2. Quartile Deviation Relative / Coefficient of Quartile Deviation (CQD) $$CQD = \frac{Q_3 - Q_1}{Q_3 + Q_1}$$

• Measures spread relative to the median range.

Differences Between Absolute and Relative Measures

Key Points

• Use absolute measures to know actual spread.
• Use relative measures to compare variability across datasets.
• Standard Deviation and CV are most commonly used.