Question

For the following data set, calculate the percentage of data points fall within one standard deviation of the mean, then compare the results to the expected percentage if a normal distribution. (55 54 66 38 53 56 57 66 45 65) percentage
Now compare the result you just got to the expected percentage if a normal distribution. The percentage expected percentage in a normal distribution.

Answer 1

First compute mean and standard deviation.

Data: 55, 54, 66, 38, 53, 56, 57, 66, 45, 65
Mean = (sum) / 10 = 555 / 10 = 55.5

Sum of squared deviations = 738.5
Population SD = sqrt(738.5/10) ≈ 8.59
Sample SD = sqrt(738.5/9) ≈ 9.06

Using either SD the ±1 SD interval is about:
- population SD: [55.5 − 8.59, 55.5 + 8.59] = [46.91, 64.09]
- sample SD: [55.5 − 9.06, 55.5 + 9.06] = [46.44, 64.56]

Data points inside that interval: 55, 54, 53, 56, 57 — 5 out of 10 points = 50%.

Comparison to a normal distribution:
- For a normal distribution about 68% (≈68.27%) of values lie within one standard deviation of the mean.
- Observed = 50% which is lower than the normal expectation (about 18 percentage points lower). This suggests the sample has fewer values near the mean (more spread or heavier tails) than a normal distribution would predict.