The Sample Variance

Defines the sample variance $S^2$ as an estimator of the population variance, motivates the divisor $n-1$ via unbiasedness and degrees of freedom, develops the computational formula in terms of $\sum X_i$ and $\sum X_i^2$ , and introduces the sample standard deviation $S = \sqrt{S^2}$ .

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Tutorial

Introducing the Sample Variance

When we draw a sample $X_1, X_2, \ldots, X_n$ from a population with unknown variance $\sigma^2$ , we estimate $\sigma^2$ from the data using the sample variance:

S^2 = \dfrac{1}{n-1}\sum\limits_{i=1}^n (X_i - \bar{X})^2,

where $\bar{X} = \dfrac{1}{n}\sum\limits_{i=1}^n X_i$ is the sample mean.

Notice the denominator: we divide by $n-1$ , not by $n$ . We will explain why in the next tutorial.

As an illustration, consider the data $\{1,\, 4,\, 7\}$ . The sample mean is

\bar{X} = \dfrac{1+4+7}{3} = 4.

The squared deviations from $\bar{X}$ are $(1-4)^2 = 9$ , $(4-4)^2 = 0$ , and $(7-4)^2 = 9$ . Their sum is $18$ , so

S^2 = \dfrac{18}{3-1} = 9.