Module 1 · Descriptive Statistics · Topic 4

Measures of Spread

Spread (Variability). Measures of spread describe how spread out the data is. Common measures include range, variance, and standard deviation.

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Key formulas

Sample Variance

s² = Σ(xᵢ − x̄)² / (n − 1)

Sample Standard Deviation

s = √[s²] = √[Σ(xᵢ − x̄)² / (n − 1)]

From the Descriptive Statistics formula sheet

Mean: xbar = sum(x_i) / n — The arithmetic average of a data set.
Sample Variance: s^2 = sum((x_i - xbar)^2) / (n - 1) — Average squared distance from the sample mean with Bessel correction.

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Worked example

Worked Example

Data: {2, 4, 4, 4, 5, 5, 7, 9}. Mean = 5. Deviations squared: 9, 1, 1, 1, 0, 0, 4, 16. Sum = 32. Sample variance = 32 / 7 ≈ 4.57. Standard deviation = √4.57 ≈ 2.14.

When to use it

Why n − 1?

Dividing by n − 1 (called Bessel’s correction) produces an unbiased estimate of the population variance. When calculating for an entire population, divide by N instead.

Related glossary terms

Outlier: A data point that is unusually far from the rest of the data. Example: In 2, 4, 5, 6, 50, the value 50 is a likely outlier.

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