68–95–99.7 rule

The 68–95–99.7 rule states that a normal distribution places about 68%, 95%, and 99.7% of observations within one, two, and three standard deviations of its mean.

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Normal distribution: A continuous probability distribution with a symmetric, bell-shaped density determined by its mean and variance. The rule describes the areas under this distribution’s curve at successive standard-deviation intervals.
Z-score: A standardized value measuring how many standard deviations an observation lies from its mean. Its magnitude identifies which of the rule’s one-, two-, or three-deviation bands contains an observation.
Chebyshev's inequality: A probability bound stating that a specified fraction of values lies near the mean for any distribution with finite variance. It applies without normality but gives weaker bounds than the rule’s normal-specific percentages.
Arithmetic mean: The sum of a set of values divided by the number of values. The rule centers every interval on the distribution’s mean.
Standard deviation: A measure of how widely values in a dataset or probability distribution vary around their mean. It sets the interval widths used to state the rule’s three percentages.
Statistical process control: Methods for monitoring whether a process remains stable by analyzing data collected over time. Control limits often use standard-deviation multiples, whose normal-model coverage the rule summarizes.
Empirical rule: A common name for the approximation that normal data fall within one, two, and three standard deviations at rates near 68%, 95%, and 99.7%. This is another name for the 68–95–99.7 rule.
Random variable: A variable whose possible values are associated with probabilities. The rule describes probabilities for values of a random variable with a normal distribution.
Standard normal distribution: A normal distribution with mean zero and standard deviation one. Standardizing values turns the rule’s intervals into the ranges from −1 to 1, −2 to 2, and −3 to 3.
Measurement uncertainty: A quantified indication of the range of values reasonably attributable to a measured quantity. When errors are modeled as normal, the rule translates uncertainty intervals into approximate coverage.