Chebyshev Theorem Calculator
Find the minimum percentage of observations within k standard deviations of the mean.
Chebyshev’s Theorem Formula
At least 1 − 1/k² of observations lie within k standard deviations of the mean for any distribution with a finite mean and variance.
How Chebyshev’s Theorem Works
Chebyshev’s theorem gives a minimum proportion of data that must lie within a chosen number of standard deviations from the mean, regardless of the distribution’s shape.
Chebyshev Example
For k = 2, at least 1 – 1/4 = 75% of observations must lie within two standard deviations of the mean. For k = 3, the minimum rises to about 88.9%.
When It Is Useful
Unlike the empirical rule, Chebyshev’s theorem does not require normally distributed data. It provides a conservative lower bound for any distribution with finite variance.
Frequently Asked Questions
Can k be less than or equal to 1?
The standard theorem is meaningful for k greater than 1.
Is the bound exact?
No. It states a guaranteed minimum, not the exact percentage.
How does it differ from the empirical rule?
The empirical rule assumes an approximately normal distribution, while Chebyshev’s theorem is distribution-independent.
Methodology and Practical Limits
Chebyshev’s theorem guarantees at least 1−1/k² of observations lie within k standard deviations of the mean for k>1, regardless of distribution shape when variance is finite. It provides a minimum bound, not an expected exact percentage.