Algebra 2 / Unit 10: Statistics and Data Analysis
10.2 Normal Distributions
Notes
A normal distribution is a symmetric, bell-shaped curve centered at the mean $\mu$ with standard deviation $\sigma$.
- Empirical rule (68-95-99.7):
- About 68% of data is within 1 standard deviation of the mean.
- About 95% is within 2 standard deviations.
- About 99.7% is within 3 standard deviations.
- Split these in half for one side: 34% between $\mu$ and $\mu + \sigma$, 47.5% between $\mu$ and $\mu + 2\sigma$.
- Example: if $\mu = 50$ and $\sigma = 10$, about 95% of values are between 30 and 70.
Practice
Heights of a group are normal with mean 64 in and standard deviation 3 in. About what percent are between 61 and 67 inches?
61 to 67 is within 1 standard deviation of the mean.
For the same heights, about what percent are taller than 70 inches?
70 is 2 standard deviations above. Outside $\pm 2\sigma$ is 5%, split evenly, so 2.5% on the high side.
For the same heights, about what percent are between 58 and 67 inches?
58 is $-2\sigma$ and 67 is $+1\sigma$: $47.5\% + 34\% = 81.5\%$.
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