Question

For the following data set, calculate the percentage of data points that fall within one standard deviation of the mean, and compare the result to the expected percentage of a normal distribution.
{50, 46, 54, 51, 29, 52, 48, 54, 47, 66}

Answer 1

Step 1 — mean:
Sum = 497, mean = 497/10 = 49.7.

Step 2 — standard deviation:
Using population formula: σ = sqrt[Σ(xi−μ)² / 10] = sqrt(762.10/10) ≈ 8.73.
(Using sample formula s = sqrt[762.10/9] ≈ 9.20 — the count below is the same for either.)

Step 3 — count within one SD:
Mean ± σ = 49.7 ± 8.73 → interval ≈ [40.97, 58.43].
Data points inside: 50, 46, 54, 51, 52, 48, 54, 47 → 8 of 10 points = 80%.

Comparison to normal expectation:
For a normal distribution about 68% of values lie within one standard deviation of the mean. This data set has 80% within one SD, which is higher than the 68% expected for a perfect normal distribution (note the sample is small and has two outliers, so conclusions about the underlying distribution are limited).