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Variance after adding the mean

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$5$ values have mean $10$ and population variance $4$ (dividing by the count). A sixth value equal to $10$ is added. What is the new population variance? Give a fraction in lowest terms.

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Hint

Track the sum of squared deviations from the mean.

Solution

The sum of squared deviations for the original values is $5 \times 4 = 20$ .
The new value equals the mean, so the mean stays $10$ and the added deviation is $0$ .
The new variance is $20 \div 6 = \frac{10}{3}$ .

Answer: $\frac{10}{3}$

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