Measures of Dispersion
Easy Overview
You and your friends score 70, 72, 68, and 71 on a test. Pretty similar, right? Now imagine scores of 30, 95, 50, and 85 — same average, but wildly different. That difference — how spread out the numbers are — is called dispersion. This chapter gives you tools to measure that spread, so you can tell if data is tightly clustered or all over the place.
Range — the simplest measure
Range = maximum value − minimum value. That's it. It's the easiest measure of dispersion. For test scores 30, 95, 50, 85, the range is 95 − 30 = 65. For 70, 72, 68, 71, the range is just 4. Quick and dirty. Downside? It only uses two numbers from the whole dataset, so outliers can mess it up badly.
Mean deviation
Mean deviation tells you 'on average, how far are the numbers from the mean?' Steps: find the mean, subtract it from each data point (take absolute value), then average those differences. For data x₁, x₂, ..., xₙ: MD = (∑|xᵢ − mean|)/n. It uses all the data, unlike range. The absolute values mean we don't care about direction — only distance.
Variance and standard deviation
This is the big one. Variance is the average of squared deviations from the mean. Instead of absolute values, we square the differences. Why square? It punishes big deviations more (like a score that's way off). For a population: σ² = ∑(xᵢ − μ)²/n. For a sample: s² = ∑(xᵢ − x̄)²/(n − 1). Standard deviation (σ or s) is just the square root of variance — it brings units back to normal. A low SD means data is clustered around the mean; a high SD means it's scattered.
Coefficient of variation
Standard deviation alone isn't always fair. Compare heights (in cm) with weights (in kg) — different units, different scales. The coefficient of variation (CV) fixes that: CV = (σ/mean) × 100%. It's a percentage. A CV of 10% means the standard deviation is 10% of the mean. The higher the CV, the more relative variability. Useful for comparing datasets with different units or vastly different means.
Key Points
- •Range = max − min — simple but sensitive to outliers
- •Mean deviation = average of |xᵢ − mean|
- •Population variance σ² = ∑(xᵢ − μ)² / n
- •Sample variance s² = ∑(xᵢ − x̄)² / (n − 1)
- •Standard deviation = √variance — in the same units as the data
- •Coefficient of variation = (σ/mean) × 100% — for comparing spreads
Practice Questions
- Find the range, variance, and standard deviation for 5, 8, 12, 15, 20.
- Two teams have the same mean score of 65. Team A has SD 8, Team B has SD 15. Which is more consistent? Why?
- Find the coefficient of variation for heights: 150, 155, 160, 165, 170 cm.
- The mean and SD of 100 items are 50 and 4. If each item is increased by 5, what happens to the mean and SD?
- Explain why variance uses squared deviations instead of absolute deviations.