Appendix A.4

Chebyshev's Theorem Calculator

Apply Chebyshev's theorem to determine the minimum percentage of data within k standard deviations.

InteractiveChebyshev's Inequality Calculator

The Formula: For any distribution, at least of observations lie within standard deviations of the mean.

Enter

(number of standard deviations):

Common values:

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

75.00%

Calculation:

💡 Key Insight: Unlike the Empirical Rule (which only applies to bell-shaped distributions), Chebyshev's Inequality works for any distribution. However, it gives a minimum guarantee, so the actual percentage is often higher.