Statistics / Unit 3: The Normal Distribution
3.2 The Normal Curve and the 68-95-99.7 Rule
Notes
A normal distribution is symmetric and bell-shaped. Its mean, median and mode are all at the center.
- 68-95-99.7 rule: about $68\%$ of values fall within 1 standard deviation of the mean, $95\%$ within 2 and $99.7\%$ within 3.
- By symmetry, about $34\%$ lie between the mean and $\mu + \sigma$, and about $2.5\%$ lie above $\mu + 2\sigma$.
- Example: Heights with $\mu = 66$ in and $\sigma = 3$ in. About $95\%$ of heights are between 60 and 72 inches.
- The rule applies only to roughly normal distributions.
Practice
Heights have a normal distribution with mean 66 inches and standard deviation 3 inches. About what percent are between 63 and 69 inches?
63 and 69 are one standard deviation below and above the mean.
For heights (mean 66, standard deviation 3), about what percent are taller than 72 inches?
72 is $\mu + 2\sigma$. $95\%$ are within 2 standard deviations, leaving $5\%$ split between two tails.
Scores are normal with mean 500 and standard deviation 100. About what percent score between 500 and 700?
500 to 700 is mean to $\mu + 2\sigma$, which is half of $95\%$.
Ready for more? Take the Unit 3 test.