Chebyshev's inequality

Chebyshev's inequality states that for any distribution, at least 1 − 1/k² of observations fall within k standard deviations of the mean, whatever the distribution's shape, even when data are not bell-shaped.

ByHoang TruongUpdated July 11, 2026

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A factory's machine cycle time has a mean of 50 seconds and a standard deviation of 4 seconds, with a distribution known to be skewed rather than normal. Taking k = 2.5 standard deviations spans 40 to 60 seconds. Chebyshev's inequality guarantees at least 1 − 1/2.5² = 84% of cycle times fall in that range, a floor that holds regardless of the distribution's actual shape.

Where it fits

TopicProbability & DistributionsAdvancedTopicDescriptive StatisticsAdvancedSubjectData Analysis & StatisticsAdvanced

The formula

LaTeX

P(|X-\mu| < k\sigma) \ge 1 - \dfrac{1}{k^2}

Variables

$X$: Random variable (an individual observation)
$μ$: Mean of the distribution
$σ$: Standard deviation of the distribution
$k$: Number of standard deviations from the mean

Gives a guaranteed minimum share of any distribution's observations lying within k standard deviations of the mean, whatever the distribution's shape.