Chebyshev's Theorem Calculator

Chebyshev's Theorem states that for any distribution — regardless of its shape — at least 1 − 1/k² of the data values will fall within k standard deviations of the mean, where k > 1. It provides a guaranteed minimum bound that applies universally, even when the distribution is not normal.

k is the number of standard deviations away from the mean that you are considering. For example, k = 2 means you are looking at the interval from (mean − 2σ) to (mean + 2σ). k must be greater than 1 for the theorem to produce a meaningful result.

When k = 1, the formula 1 − 1/k² equals zero, which gives no useful information (it only guarantees 0% of data is within 1 standard deviation). For k less than or equal to 1, the result would be zero or negative, which is meaningless. The theorem only provides useful bounds for k > 1.

The formula is: Minimum proportion within k std devs = 1 − 1/k². Multiply by 100 to get the percentage. For example, with k = 2: 1 − 1/4 = 0.75, so at least 75% of data lies within 2 standard deviations of the mean.

The Empirical Rule (68-95-99.7 rule) applies only to normally distributed data and gives exact percentages (e.g. exactly 95% within 2 standard deviations). Chebyshev's Theorem works for any distribution but gives a minimum guarantee — for k = 2, it guarantees at least 75%, whereas the Empirical Rule specifies 95% for a normal distribution.

Yes — that is its greatest strength. Unlike the Empirical Rule, Chebyshev's Theorem requires no assumptions about the shape of the distribution. It applies to skewed data, bimodal data, uniform distributions, and any other shape, making it a very broadly applicable statistical tool.

This is the complement: at most 1/k² of the data can lie outside the range of k standard deviations from the mean. For k = 2, that means no more than 25% of the data falls beyond 2 standard deviations. It gives you an upper bound on the proportion of outliers.

For k = 1.5, at least 55.56% of data falls within 1.5 standard deviations. For k = 2, at least 75%. For k = 3, at least 88.89%. For k = 4, at least 93.75%. The larger k is, the higher the guaranteed minimum percentage within that range.