The 68-95-99.7 Rule

The Empirical Rule

For an exactly normal variable, fixed proportions lie near the mean:

about 68.27% within $μ \pm σ$
about 95.45% within $μ \pm 2 σ$
about 99.73% within $μ \pm 3 σ$

The popular 68-95-99.7 values are rounded approximations to Gaussian integrals. For data that is only approximately normal, the proportions are approximations twice over.

Do Not Confuse Three Different Intervals

A data interval such as $μ \pm 2 σ$ describes individual values under a normal model.
A confidence interval describes uncertainty in an estimated parameter and usually depends on a standard error, sample size, and sampling assumptions.
A p-value is a tail probability for a test statistic under a null hypothesis; “p<0.05” does not follow from the empirical rule by itself.

A value beyond 3σ may deserve investigation, but calling it an error also requires domain knowledge, multiple-testing awareness, and a credible distributional model.

The empirical rule describes observations from a normal distribution. It is not a shortcut that automatically creates confidence intervals or significance tests.

Code

Empirical verification of the rule·python

import numpy as np

samples = np.random.normal(0, 1, 100_000)
within_1_sigma = np.mean(np.abs(samples) < 1)
within_2_sigma = np.mean(np.abs(samples) < 2)
within_3_sigma = np.mean(np.abs(samples) < 3)
print(f"|x| < 1σ: {within_1_sigma:.3f}")  # ~0.683
print(f"|x| < 2σ: {within_2_sigma:.3f}")  # ~0.954
print(f"|x| < 3σ: {within_3_sigma:.3f}")  # ~0.997

Exercise

If IQs are normally distributed with mean 100 and std 15, roughly what fraction of people have an IQ above 130? What about above 145?

Hint

130 = mean + 2σ → about (100% - 95%)/2 ≈ 2.5% above. 145 = mean + 3σ → about (100% - 99.7%)/2 ≈ 0.15% above. Two-sigma and three-sigma cutoffs in the wild.

The Empirical Rule

Do Not Confuse Three Different Intervals

Code

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Exercise

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