If the following data is obtained from a simple random sample:
6, 7, 9, 10, 11, 17
Then the point estimate of population standard deviation is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3.898
The point estimate of the population standard deviation is the sample standard deviation s with divisor n − 1, since \(s^2\) with divisor n − 1 is the unbiased estimate of the population variance. Here n = 6:
\( \displaystyle \bar{x} = \frac{6 + 7 + 9 + 10 + 11 + 17}{6} = \frac{60}{6} = 10 \)
The squared deviations from the mean are 16, 9, 1, 0, 1 and 49:
\(\sum (x - \bar{x})^2 = 16 + 9 + 1 + 0 + 1 + 49 = 76\)
\( \displaystyle s^2 = \frac{76}{6 - 1} = 15.2 \)
\( \displaystyle s = \sqrt{15.2} = 3.8987\ldots \)
The option gives this value as 3.898.
Why the other options are incorrect
Option (1) 15.21 — Incorrect. This is about the variance \(s^2 = 15.2\), not the standard deviation.
Option (2) 10 — Incorrect. This is the sample mean.
Option (4) 3.192 — Incorrect. Neither divisor gives it: n − 1 gives 3.898, and n gives \(\sqrt{76/6} = 3.559\).