The 68-95-99.7 Rule

~6 min · empirical-rule, standard-deviation

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One Magic Sentence

For a normal distribution, fixed proportions of the data lie within 1, 2, or 3 standard deviations of the mean:

~68% within $μ \pm σ$
~95% within $μ \pm 2 σ$
~99.7% within $μ \pm 3 σ$

This is the empirical rule — sometimes called 68-95-99.7. It's not approximate; it's the actual integral of the Gaussian density at those bounds.

Practical Use

Outlier detection: anything past 3σ is rare enough to investigate (~0.3%).
Confidence intervals: ±2σ is roughly the 95% interval — basis of "p < 0.05" significance testing.
Six Sigma quality: defects past 6σ from the target — extremely rare events. The name of the methodology comes directly from this rule.

If a distribution is approximately normal, you know how spread out the data is the moment you know μ and σ. The 68-95-99.7 rule is the cheat code.

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

External links

68-95-99.7 rule — Wikipedia

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.

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