Sample variance

Let ${𝑋 𝑗} 𝑛 𝑗 = 1$ be a sample of independent and identically distributed real random variables with mean $𝜇$ and variance $𝜎 2$ . The sample variance is #m/def/prob

𝑆 2 𝑛 = 1 𝑛 - 1 𝑛 \sum 𝑗 = 1 (𝑋 𝑗 - ―― 𝑋 𝑛) 2

where $―― 𝑋 𝑗$ is the sample mean. $𝑆 2 𝑛$ estimates the variance of the population since

𝔼 [𝑆 2 𝑛] = 𝜎 2

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