The following data is taken from a simple random sample : 3, 7, 5, 9,…

The correct answer is Option (3) → \(\displaystyle\frac{2}{3}\sqrt{35}\)

The point estimate of the population standard deviation is the sample standard deviation with divisor \(n - 1\):

    \( \displaystyle s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n-1}} \)

Here \(n = 10\) and

    \( \displaystyle \bar{x} = \frac{3 + 7 + 5 + 9 + 15 + 11 + 8 + 4 + 6 + 2}{10} = \frac{70}{10} = 7 \)

The deviations from 7 are −4, 0, −2, 2, 8, 4, 1, −3, −1, −5, so

    \(\sum (x - \bar{x})^{2} = 16 + 0 + 4 + 4 + 64 + 16 + 1 + 9 + 1 + 25 = 140\)

    \( \displaystyle s = \sqrt{\frac{140}{9}} = \frac{\sqrt{4 \times 35}}{3} = \frac{2}{3}\sqrt{35} \)

Tip: Dividing by n instead gives \(\sqrt{14}\), the standard deviation of the sample itself. The point estimate of the population standard deviation always uses \(n - 1\).