Question

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 48) Percentage
The percentage than the expected percentage of a normal distribution.

Answer 1

Step 1 — compute mean
Sum = 479, n = 10 → mean = 479/10 = 47.9

Step 2 — compute standard deviation (sample s)
Sum of squared deviations = 466.90
s^2 = 466.90/(10−1) = 51.8778 → s ≈ 7.21

Interval mean ± 1 s:
47.9 ± 7.21 → [40.69, 55.11]

Step 3 — count points inside that interval
Data inside: 50, 46, 54, 51, 52, 48, 54, 47, 48 = 9 of 10 → 90%

Comparison with normal distribution
Expected percentage within ±1σ for a normal ≈ 68.27%. Your sample: 90%, which is 90% − 68.27% ≈ 21.73 percentage points higher (about 1.32 times the normal expectation).

Interpretation: In this small sample, a larger fraction of observations lie near the mean than a normal distribution would predict.