Chebyshev's Theorem Calculator

Chebyshev's theorem gives a guaranteed lower bound for the share of observations located near the mean of a distribution. Unlike the familiar empirical rule, it does not require the data to follow a normal, symmetric, or bell-shaped distribution. If a population or data set has a finite mean and finite variance, at least 1 − 1/k² of its values must lie within k standard deviations of the mean for every k greater than one. This broad applicability makes the theorem useful when the shape of a distribution is unknown, irregular, or strongly skewed. The calculator applies two connected formulas. First, it computes the interval from μ − kσ to μ + kσ, where μ is the mean, σ is the standard deviation, and k is the selected distance. Second, it converts 1 − 1/k² to a percentage. For k = 2, the theorem guarantees at least 75% of values within two standard deviations. For k = 3, the guarantee rises to about 88.89%. These are minimum bounds, not predictions of the exact percentage. A particular distribution may place considerably more data inside the interval. Chebyshev bounds are conservative because they must remain valid for almost any distribution. For normally distributed data, approximately 95% of observations fall within two standard deviations, while Chebyshev guarantees only 75%. That difference does not make the theorem inaccurate. It reflects the strength of making no assumption about distribution shape. When normality is justified, distribution-specific methods provide tighter estimates; when it is not, Chebyshev offers a dependable baseline. The interval uses the same units as the original measurements. A mean test score of 70 with a standard deviation of 8 and k = 2 produces an interval from 54 to 86 and guarantees at least 75% of scores in that range. The result concerns the proportion of the full distribution, not the probability of an individual future outcome unless the observation is randomly drawn from that distribution. Always confirm that the mean and standard deviation describe the population or sample you intend to analyze, and remember that k must exceed one for the bound to be informative.