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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.
{55, 54, 66, 38, 53, 56, 57, 66, 45, 65}
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Mean = 55.5.
Using the population standard deviation:
variance = (sum of (xi − mean)^2)/10 = 738.5/10 = 73.85 → SD ≈ 8.59.
One-SD interval = 55.5 ± 8.59 = [46.91, 64.09].
Data points in that interval: 55, 54, 53, 56, 57 → 5 out of 10 = 50%.
(If you use the sample SD you get SD ≈ 9.06 and interval [46.44, 64.56], which still contains the same 5 points → 50%.)
Comparison: for a normal distribution about 68% of values fall within one standard deviation of the mean, so this sample (50%) has fewer points within one SD than the normal expectation — the data are not closely following the empirical rule here.
Using the population standard deviation:
variance = (sum of (xi − mean)^2)/10 = 738.5/10 = 73.85 → SD ≈ 8.59.
One-SD interval = 55.5 ± 8.59 = [46.91, 64.09].
Data points in that interval: 55, 54, 53, 56, 57 → 5 out of 10 = 50%.
(If you use the sample SD you get SD ≈ 9.06 and interval [46.44, 64.56], which still contains the same 5 points → 50%.)
Comparison: for a normal distribution about 68% of values fall within one standard deviation of the mean, so this sample (50%) has fewer points within one SD than the normal expectation — the data are not closely following the empirical rule here.
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