Module 1 · Descriptive Statistics · Topic 4
Spread (Variability). Measures of spread describe how spread out the data is. Common measures include range, variance, and standard deviation.
s² = Σ(xᵢ − x̄)² / (n − 1)
s = √[s²] = √[Σ(xᵢ − x̄)² / (n − 1)]
xbar = sum(x_i) / n - The arithmetic average of a data set.s^2 = sum((x_i - xbar)^2) / (n - 1) - Average squared distance from the sample mean with Bessel correction.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.
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.
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