Chebyshevs Theorem Calculator

Understanding Chebyshev's Theorem (Chebyshev's Inequality)

Chebyshev's Theorem (also known as Chebyshev's Inequality) is a fundamental rule in probability and statistics. It guarantees that for ANY data distribution (regardless of shape, skewness, or symmetry), at least a minimum fraction of data points lie within $k$ standard deviations of the mean.

Chebyshev's Formula

For any real number $k > 1$, the minimum proportion of data within $k$ standard deviations ($\mu \pm k\sigma$) is:

$$P(\mu - k\sigma < X < \mu + k\sigma) \ge 1 - \frac{1}{k^2}$$

Common Values of K

• For $k = 2$: At least $1 - \frac{1}{4} = 75\%$ of data lies within $2$ standard deviations.
• For $k = 3$: At least $1 - \frac{1}{9} \approx 88.89\%$ of data lies within $3$ standard deviations.
• For $k = 4$: At least $1 - \frac{1}{16} = 93.75\%$ of data lies within $4$ standard deviations.

Chebyshev's Theorem vs. The Empirical Rule

While the Empirical Rule ($68\%-95\%-99.7\%$) applies strictly to bell-shaped normal distributions, Chebyshev's Theorem works universally for any dataset, including skewed, bimodal, or non-normal data distributions.

Explore non-parametric data summaries with our Five-Number Summary Calculator, or smooth out time series data using the Moving Average Calculator.