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Variance of a sample mean

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A population consists of the numbers $1, 2, 3, 4, 5, 6$ . You draw $3$ of them without replacement, with all subsets equally likely, and take the mean of the drawn numbers. What is the variance of this sample mean?

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Hint

Start from the population variance, divided by the population size, then apply the finite-population correction.

Solution

The population variance, dividing by $6$ , is $σ^{2} = \frac{35}{12}$ .
Without replacement, $Var (mean) = \frac{σ ^{2}}{n} \times \frac{N - n}{N - 1}$ .
This gives $\frac{35}{36} \times \frac{3}{5} = \frac{7}{12}$ .

Answer: $\frac{7}{12}$

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