Chebyshev's Theorem Calculator

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Forward mode: enter k standard deviations to get the minimum percentage guaranteed within that range. Reverse mode: enter a target percentage to find the minimum k required.
Must be greater than 1. Chebyshev's theorem is trivially satisfied for k <= 1 and gives no useful bound.
Optional. Enter the distribution mean to display the actual interval [mean - k*sd, mean + k*sd] alongside the proportion.
Optional. Enter the standard deviation to display the actual interval in original units.
Minimum proportion within rangeModerate guarantee (>= 75%)
75%

At least this percentage of data lies within k standard deviations of the mean

Required k2
Maximum proportion outside range0.25%
Lower interval bound70
Upper interval bound130
Interval[70.0000, 130.0000]
75% proportion
Weak (<75%)<75%Moderate (75-88.9%)75%-88.9%Strong (88.9-96%)88.9%-96%Very strong (>96%)96%+

At least 75.00% of any distribution lies within 2.0000 standard deviations of the mean.

  • Chebyshev's theorem guarantees that no more than 25.00% of values fall outside this symmetric interval around the mean.
  • This bound applies to every distribution with a finite mean and variance - normal, skewed, bimodal, or unknown shape.
  • For k = 2, the Empirical Rule guarantees 95% for normal distributions, but Chebyshev only guarantees 75% for arbitrary distributions. The gap reflects how much the normal shape helps.

Next stepIf you know your data is approximately normal, use the Empirical Rule (68-95-99.7%) for tighter estimates. Use Chebyshev's theorem when the distribution is unknown or non-normal.

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