Integrated Math 3 / Unit 8: Statistics and Inference
8.1 The Normal Distribution and the Empirical Rule
Notes
A normal distribution is symmetric and bell-shaped, described by its mean $\mu$ and standard deviation $\sigma$.
- Empirical Rule (68-95-99.7): about 68% of values are within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
- The mean, median and mode are all at the center.
- Example: with $\mu = 70$ and $\sigma = 5$, about 95% of values are between 60 and 80.
- Because of symmetry, about 16% of values are more than 1 standard deviation above the mean ($\displaystyle \frac{100 - 68}{2}$).
Practice
Test scores are normal with mean 70 and standard deviation 5. About what percent of scores are between 60 and 80?
60 and 80 are 2 standard deviations below and above the mean.
Test scores are normal with mean 70 and standard deviation 5. About what percent of scores are above 75?
75 is 1 standard deviation above the mean. Outside $\pm 1$ standard deviation is 32%, and half of that is above.
Test scores are normal with mean 70 and standard deviation 5. About what percent of scores are between 65 and 80?
65 is $-1\sigma$ and 80 is $+2\sigma$. That is $34\% + 47.5\% = 81.5\%$.
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