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68–95–99.7 rule

Also called: empirical rule, three-sigma rule

For a normal curve, about 68% of values lie within 1 SD of the mean, 95% within 2 SD (exactly ±1.96) and 99.7% within 3 SD.

Under a normal curve, area is probability: the share of the area between two points is the share of values there, and the whole area is 1. Three areas are worth knowing by heart:

  • within ±1 SD of the mean: about 68%;
  • within ±2 SD: about 95%;
  • within ±3 SD: about 99.7%.

The rest splits equally between the two tails, because the curve is symmetric: 2.5% above +2 SD and 2.5% below −2 SD.

The "2" is rounded. The exact z that leaves 95% in the middle is 1.96, which is why 95% confidence intervals are x̄ ± 1.96 × SE and two-tailed tests at 5% compare |z| with 1.96. To use the rule on any data, turn a value into z = (x − mean) ÷ SD first.

Example

Mock exam scores have mean 150 and SD 15.

  • 68% score between 135 and 165.
  • 95% score between 120 and 180, so about 2.5% score above 180.
  • 99.7% score between 105 and 195.

Common mistakes

  • Applying it to skewed data. Revenue per user or session length are not normal; averages of many of them are closer.
  • Forgetting the tails split. 5% outside ±2 SD is 2.5% on each side.
  • Using 2 where 1.96 is meant. Fine for a quick look, not for an interval you report.
  • Reading width as area. The bands are shares of the area, not of the axis.

Learn it in the course

  • The normal curve · Area as probability, the 68–95–99.7 rule, where 1.96 comes from, one- and two-tailed p-values, and the z-values for α and power.