Empirical Rule Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Empirical Rule.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The empirical rule (also called the 68-95-99.7 rule) states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

Most data clusters near the center of a bell curve; the further from the mean, the rarer the value.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: For a roughly normal distribution, the empirical rule gives the percentage of data inside one, two, and three standard deviations of the mean: about 68%, 95%, and 99.7%.

Common stuck point: Students often know a procedure related to empirical rule but skip the recognition step: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I reasoning about what can happen and how likely it is, with the correct sample space or condition?

Worked Examples

Example 1

medium
Heights of adult men are normal with μ=70\mu = 70 in, σ=3\sigma = 3 in. Estimate the proportion of men taller than 7676 in.

Answer

≈2.5%\approx 2.5\%

First step

1
76=70+2⋅3=μ+2σ76 = 70 + 2 \cdot 3 = \mu + 2\sigma.

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Example 2

medium
In a normal distribution, what percent lies between μ−σ\mu - \sigma and μ+2σ\mu + 2\sigma?

Example 3

hard
A normal distribution has μ=500\mu = 500, σ=100\sigma = 100. About what percent lies between 400400 and 700700?

Example 4

hard
A factory makes bolts with normal length, μ=50\mu = 50 mm, σ=0.5\sigma = 0.5 mm. Bolts shorter than 4949 mm are rejected. About what fraction is rejected?

Example 5

hard
Heights of women are normal with μ=64\mu = 64 in, σ=3\sigma = 3 in. About how many women out of 10001000 are between 5858 and 7070 inches tall?

Example 6

challenge
A factory produces tablets whose weights are normal with μ=500\mu = 500 mg and σ=8\sigma = 8 mg. Tablets weighing less than 484484 mg or more than 516516 mg are rejected. Approximately what fraction of tablets pass inspection?

Example 7

hard
IQ scores are normal, μ=100\mu = 100, σ=15\sigma = 15. About what percent of people have IQ between 7070 and 145145?

Example 8

challenge
For a standard normal distribution, estimate the percent of data between −1.5-1.5 and +1.5+1.5 standard deviations using linear interpolation between the 11-SD and 22-SD empirical landmarks, then compare to the true value.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
State the three percentages of the empirical rule.

Example 2

easy
About what percent of normal data lies within 22 standard deviations of the mean?

Example 3

easy
About what percent of normal data lies within 33 standard deviations of the mean?

Example 4

easy
Does the empirical rule apply to a heavily skewed distribution?

Example 5

easy
In a normal distribution, what percent of data lies OUTSIDE 22 standard deviations?

Example 6

easy
Are the empirical-rule percentages exact or approximate?

Example 7

easy
In a normal distribution, what percent lies BELOW the mean?

Example 8

easy
What percent of normal data lies between the mean and +1+1 standard deviation?

Example 9

medium
IQ scores are normal with μ=100\mu=100, σ=15\sigma=15. What percent of people have IQ between 8585 and 115115?

Example 10

medium
IQ is normal, μ=100\mu=100, σ=15\sigma=15. What percent of people have IQ above 130130?

Example 11

medium
A normal distribution has μ=200\mu=200, σ=20\sigma=20. What percent of data lies between 160160 and 240240?

Example 12

medium
Heights are normal, μ=170\mu=170, σ=10\sigma=10. What percent of people are between 150150 and 190190 cm?

Example 13

medium
A normal distribution, μ=50\mu=50, σ=10\sigma=10. What percent lies between 5050 and 7070?

Example 14

medium
A normal distribution, μ=80\mu=80, σ=6\sigma=6. What percent of data lies below 7474?

Example 15

medium
Why does the empirical rule fail for the time between bus arrivals (often right-skewed)?

Example 16

medium
A normal distribution, μ=100\mu=100, σ=10\sigma=10. What percent lies between 9090 and 120120?

Example 17

medium
A normal distribution has μ=70\mu=70, σ=5\sigma=5. What percent of data lies above 8080?

Example 18

challenge
Scores are normal with μ=500\mu=500, σ=100\sigma=100. Estimate the percent scoring between 300300 and 600600.

Example 19

challenge
In a normal distribution, what percent of data lies between +1σ+1\sigma and +2σ+2\sigma above the mean?

Example 20

challenge
A factory's part lengths are normal with μ=50\mu=50 mm, σ=2\sigma=2 mm. Parts outside 4646 to 5454 mm are scrapped. What percent is scrapped?

Example 21

easy
For a normal distribution, what percent of data lies between μ−σ\mu - \sigma and μ+σ\mu + \sigma?

Example 22

easy
A normal distribution has μ=50\mu = 50, σ=5\sigma = 5. About what percent of values fall between 4545 and 5555?

Example 23

easy
In a normal distribution, what percent lies outside ±3\pm 3 SD?

Example 24

easy
For a normal distribution, what percent of data lies below μ−2σ\mu - 2\sigma?

Example 25

easy
In a normal distribution with μ=0\mu = 0, σ=1\sigma = 1, what percent lies between −3-3 and 33?

Example 26

medium
SAT scores are normal with μ=1000\mu = 1000, σ=200\sigma = 200. What percent of test-takers score between 800800 and 12001200?

Example 27

medium
SAT scores are normal with μ=1000\mu = 1000, σ=200\sigma = 200. What percent score above 14001400?

Example 28

medium
A test is normal with μ=75\mu = 75, σ=10\sigma = 10. What range covers about 95%95\% of scores?

Example 29

medium
A normal distribution has μ=100\mu = 100, σ=10\sigma = 10. What percent of data lies between 9090 and 110110?

Example 30

medium
Normal: μ=60\mu = 60, σ=4\sigma = 4. About what percent of values lie below 5252?

Example 31

hard
In a normal distribution with μ=50\mu = 50, σ=4\sigma = 4, what percent of values lie between 4242 and 5858?

Example 32

hard
Normal: μ=200\mu = 200, σ=25\sigma = 25. What percent of values lie above 250250?

Example 33

hard
In a normal distribution, what percent of values lie between μ+σ\mu + \sigma and μ+2σ\mu + 2\sigma?

Example 34

hard
A normal distribution has μ=0\mu = 0, σ=1\sigma = 1. About what percent of values lie above 11?

Example 35

medium
A normal distribution has μ=20\mu = 20, σ=3\sigma = 3. What percent of data lies between 1414 and 2323?
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Related Concepts

Normal Distribution

Background Knowledge

These ideas may be useful before you work through the harder examples.

stat normal distribution