Enter a value for k (the number of standard deviations from the mean) and this Chebyshev's Theorem Calculator tells you the minimum percentage of data that falls within that range — for any distribution shape. Works with any dataset, no normality assumption required.
Minimum % of Data Within k Standard Deviations
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Maximum % of Data Outside k Standard Deviations
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k² Value
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Minimum Probability (Decimal)
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Chebyshev's Theorem states that for any dataset — regardless of its distribution shape — at least (1 − 1/k²) × 100% of data values will fall within k standard deviations of the mean. This makes it a powerful, distribution-free rule applicable to any dataset.
When k equals 1 or less, the formula 1 − 1/k² produces zero or a negative result, which provides no meaningful probability bound. Chebyshev's Theorem is only useful for k values strictly greater than 1.
The Empirical Rule (68-95-99.7 rule) applies only to normal (bell-shaped) distributions. Chebyshev's Theorem works for any distribution shape, but provides a more conservative (lower) minimum percentage than the Empirical Rule for the same k value.
The result is the minimum guaranteed percentage of data that lies within k standard deviations of the mean. For example, if the result is 75%, at least 75% of your data points fall between (mean − k×SD) and (mean + k×SD), no matter the shape of your distribution.
With k = 2, the theorem guarantees at least 1 − 1/4 = 75% of data falls within 2 standard deviations of the mean. Compare this to the Empirical Rule's 95% for normal distributions — the difference reflects Chebyshev's universal applicability.
With k = 3, Chebyshev's Theorem guarantees at least 1 − 1/9 ≈ 88.89% of data falls within 3 standard deviations. The Empirical Rule gives 99.7% for normal distributions, but Chebyshev's result holds for any shape.
Yes — that's one of its biggest strengths. Chebyshev's Theorem makes no assumptions about distribution shape. Whether your data is skewed, bimodal, or follows any other pattern, the theorem's probability bound still holds.
The calculator uses the formula: P ≥ 1 − (1 / k²), where k is the number of standard deviations. The result is multiplied by 100 to express it as a percentage. The maximum percentage outside is simply 100% minus this result.