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Statistics

Empirical Rule (68-95-99.7)

Lesson

For any normal distribution, the percentage of data within a given number of standard deviations of the mean is almost always the same. This is the empirical rule (also called the 68-95-99.7 rule):

  • About 68% of the data lies within 1 standard deviation of the mean.
  • About 95% within 2 standard deviations.
  • About 99.7% within 3 standard deviations.

Combine this with the symmetry of the bell curve (50% above the mean, 50% below) and you can read off lots of percentages by adding pieces:

  • Mean to μ+σ\mu + \sigma: 34% (half of 68).
  • μ+σ\mu + \sigma to μ+2σ\mu + 2\sigma: 13.5% ((95 − 68)/2).
  • μ+2σ\mu + 2\sigma to μ+3σ\mu + 3\sigma: 2.35% ((99.7 − 95)/2).
  • Beyond μ+3σ\mu + 3\sigma: 0.15% ((100 − 99.7)/2).
  • Same pattern on the left side by symmetry.

Worked example 1

Test scores are normal with μ=100, σ=15\mu = 100,\ \sigma = 15. What percent of students score between 85 and 115?

That range is exactly μ−σ\mu - \sigma to μ+σ\mu + \sigma — within 1 standard deviation:

68%68\%

Worked example 2 — adding pieces

Same distribution. What percent score above 85?

85 = μ−σ\mu - \sigma. “Above 85” = from μ−σ\mu - \sigma up to +∞+\infty. That’s the 34% from μ−σ\mu - \sigma to μ\mu, plus the 50% above the mean:

34%+50%=84%34\% + 50\% = 84\%

How to type your answer

Type the percent without the % sign. For sixty-eight percent, type 68. Examples: 68, 95, 13.5, 0.15.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

μ=100, σ=15. % between 85 and 115\mu = 100,\ \sigma = 15.\ \%\text{ between 85 and 115}

Problem 2

μ=100, σ=15. % between 70 and 130\mu = 100,\ \sigma = 15.\ \%\text{ between 70 and 130}

Problem 3

μ=100, σ=15. % between 55 and 145\mu = 100,\ \sigma = 15.\ \%\text{ between 55 and 145}

Problem 4

μ=50, σ=10. % between 40 and 60\mu = 50,\ \sigma = 10.\ \%\text{ between 40 and 60}

Practice

Standard problems matching the lesson.

Problem 5

μ=50, σ=10. % above 60\mu = 50,\ \sigma = 10.\ \%\text{ above 60}

Problem 6

μ=50, σ=10. % below 40\mu = 50,\ \sigma = 10.\ \%\text{ below 40}

Problem 7

μ=100, σ=15. % above 115\mu = 100,\ \sigma = 15.\ \%\text{ above 115}

Problem 8

μ=100, σ=15. % below 70\mu = 100,\ \sigma = 15.\ \%\text{ below 70}

Problem 9

μ=100, σ=15. % between 100 and 130\mu = 100,\ \sigma = 15.\ \%\text{ between 100 and 130}

Problem 10

μ=80, σ=8. % between 80 and 96\mu = 80,\ \sigma = 8.\ \%\text{ between 80 and 96}

Problem 11

μ=80, σ=8. % above 96\mu = 80,\ \sigma = 8.\ \%\text{ above 96}

Problem 12

μ=70, σ=12. % between 70 and 94\mu = 70,\ \sigma = 12.\ \%\text{ between 70 and 94}

Problem 13

μ=60, σ=5. % between 50 and 70\mu = 60,\ \sigma = 5.\ \%\text{ between 50 and 70}

Problem 14

μ=60, σ=5. % between 55 and 65\mu = 60,\ \sigma = 5.\ \%\text{ between 55 and 65}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

μ=100, σ=15. % between 115 and 130\mu = 100,\ \sigma = 15.\ \%\text{ between 115 and 130}

Problem 16

μ=100, σ=15. % below 70\mu = 100,\ \sigma = 15.\ \%\text{ below 70}

Problem 17

μ=100, σ=15. % above 145\mu = 100,\ \sigma = 15.\ \%\text{ above 145}

Problem 18

μ=100, σ=15. % between 85 and 130\mu = 100,\ \sigma = 15.\ \%\text{ between 85 and 130}

Problem 19

μ=100, σ=15. % outside the range 85 to 115\mu = 100,\ \sigma = 15.\ \%\text{ outside the range 85 to 115}

Problem 20

μ=100, σ=15. % between 55 and 70\mu = 100,\ \sigma = 15.\ \%\text{ between 55 and 70}

Problem 21

μ=100, σ=15. % within 3 SDs of the mean\mu = 100,\ \sigma = 15.\ \%\text{ within 3 SDs of the mean}

Problem 22

μ=100, σ=15. % above 85\mu = 100,\ \sigma = 15.\ \%\text{ above 85}