Chebyshev's Theorem Calculator
Chebyshev's inequality applies to any distribution with a well-defined mean and variance. Enter k to obtain the guaranteed minimum coverage within ±k standard deviations.
Must be greater than 1 for a meaningful bound.
Minimum % Within ±kσ
75.00%
Guaranteed coverage
Maximum % Outside
25.00%
Upper bound on tail probability
Interval
[-2.00, 2.00]
Relative to the mean
How to Use This Calculator
- Enter the number of standard deviations (k) from the mean you are interested in.
- Optionally provide the distribution's mean and standard deviation to see the numeric interval.
- Read the guaranteed minimum proportion of observations that Chebyshev's theorem ensures will fall within that interval.
- Use the outside bound to understand the maximum possible tail probability.
Formula
Rearranging gives P(|X − μ| < kσ) ≥ 1 − 1/k². The result holds for any distribution with finite variance, regardless of shape.
Full Description
Chebyshev's inequality is a fundamental result in probability theory. It guarantees that for any random variable with mean μ and standard deviation σ, at least 1 − 1/k² of the distribution falls within k standard deviations of the mean. Unlike the empirical rule (68-95-99.7%), Chebyshev does not assume normality, so the bound can be conservative but universally valid.
Analysts use Chebyshev's theorem when little is known about the underlying distribution. It provides a safety net: no matter how heavy-tailed or skewed the data is, the theorem still bounds how much mass can lie far from the mean.
Frequently Asked Questions
Is Chebyshev's bound tight?
The bound can be achieved in some extreme distributions, but it is often conservative in practice. For well-behaved distributions (like normal), the actual coverage is much higher.
Can k be less than 1?
The inequality holds for any positive k, but it only provides non-trivial information when k > 1. Below that threshold, the bound becomes greater than or equal to 1.
How does this compare to the Empirical Rule?
The empirical rule (68-95-99.7) relies on normality assumptions and gives tighter estimates. Chebyshev is broader and applies even when the distribution is not normal.
Can I use Chebyshev for sample data?
Yes. Estimate the sample mean and standard deviation to approximate the bound. The inequality still offers a conservative guarantee about the underlying population.