Question

calculate the standard deviation given these raw scores
48, 54, 39, 57, 40, 44

Explanation:

Step1: Calculate the mean

First, sum all scores then divide by the count.
$\bar{x} = \frac{48 + 54 + 39 + 57 + 40 + 44}{6} = \frac{282}{6} = 47$

Step2: Find squared deviations

Subtract mean from each score, square the result.
$(48-47)^2=1, (54-47)^2=49, (39-47)^2=64, (57-47)^2=100, (40-47)^2=49, (44-47)^2=9$

Step3: Compute variance (sample)

Sum squared deviations, divide by $n-1$.
$s^2 = \frac{1 + 49 + 64 + 100 + 49 + 9}{6-1} = \frac{272}{5} = 54.4$

Step4: Calculate standard deviation

Take square root of variance.
$s = \sqrt{54.4} \approx 7.376$

Answer:

$\approx 7.38$ (or exact form $\sqrt{54.4}$)