The empirical rule • 68% · 95% · 99.7% • Read a bell curve by standard deviations
A normal distribution is a bell-shaped, symmetric curve. Most values sit near the mean. The empirical rule tells you about what percent of the data fall within 1, 2, or 3 standard deviations of that mean.
About 68%
of the data are within 1 standard deviation of the mean.
About 95%
of the data are within 2 standard deviations of the mean.
About 99.7%
of the data are within 3 standard deviations of the mean.
How to use it
Find the mean. Count how many standard deviations the endpoints are from the mean. Then pick 68%, 95%, or 99.7%. Because the curve is symmetric, each half of the 68% band is about 34%.
The empirical-rule graph
Read the arrows: 68% sits between −1 and +1, 95% sits between −2 and +2, and 99.7% sits between −3 and +3.
In a normal distribution, about what percent of the data fall within 3 standard deviations of the mean?
The empirical rule says about 99.7% of the data in a normal distribution fall within 3 standard deviations of the mean (from −3 to +3 on the graph).
Answer: C
Test scores are approximately normal with mean 80 and standard deviation 10. About what percent of the scores are between 70 and 90?
70 is 80 − 10, and 90 is 80 + 10. Those endpoints are 1 standard deviation from the mean, so the empirical rule gives about 68%.
Answer: A
The same scores have mean 80 and standard deviation 10. About what percent of the scores are between 60 and 100?
60 is 80 − 2(10), and 100 is 80 + 2(10). Those endpoints are 2 standard deviations from the mean, so the empirical rule gives about 95%.
Answer: B
Adult heights in a group are approximately normal with mean 66 inches and standard deviation 3 inches. About 95% of the heights fall between which two values?
95% is the band 2 standard deviations from the mean.
66 − 2(3) = 60 and 66 + 2(3) = 72.
A is only 1 standard deviation (68%). C is 3 standard deviations (99.7%).
Answer: B
Quiz scores are approximately normal with mean 75 and standard deviation 8. About what percent of the scores are greater than 83?
83 is 75 + 8, so it is 1 standard deviation above the mean.
About 68% of scores sit between 67 and 83. That leaves 32% outside that band, 16% in each tail. The upper tail is the part greater than 83.
34% is the piece from the mean up to 83, not the tail above 83.
Answer: A
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