A height within three standard deviations

Question

Heights are normally distributed with a mean of 160 cm and a standard deviation of 8 cm. Use the empirical (68-95-99.7) rule.

What is the probability that a randomly chosen height falls between 136 and 184 cm?

Accepted Answer

The empirical rule in standard-deviation steps: - about 68% of values fall within 1 standard deviation of the mean, - about 95% within 2 standard deviations, - about 99.7% within 3 standard deviations. Measure the endpoints from the mean of 160 cm with a spread of 8 cm. The value 136 cm is \frac{136 - 160}{8} = -3 standard deviations, and 184 cm is \frac{184 - 160}{8} = +3 standard deviations. So the band from 136 to 184 cm is exactly "within 3 standard deviations of the mean." By the rule, that band holds about 99.7% of all heights: P(136 \le X \le 184) \approx 0.997. Almost everyone falls i…

A height within three standard deviations

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