Chebyshev's Inequality Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
Chebyshev's inequality guarantees that at least (1 − 1/k²) × 100% of the data lies within k standard deviations of the mean, regardless of the shape of the distribution. At k=2 standard deviations, at least 75% of the data is guaranteed to fall within that interval.
Explanation
Standard deviations from the mean are also the basis of the z-score calculator already published on this site, which locates a single value relative to a distribution rather than bounding a whole proportion of data — but for a distribution known to be normal (bell-shaped), the exact proportions within 1, 2 or 3 standard deviations of the mean are much higher than what Chebyshev guarantees. Chebyshev's inequality answers a different, more general question: what can be guaranteed about any distribution, even a completely unknown or clearly asymmetric one, as long as it has a finite mean and standard deviation? The answer is a minimum bound, not an exact value: at least 1 − 1/k² of all the data lies within k standard deviations of the mean, with no assumption whatsoever about the shape of the distribution. This guarantee is necessarily weaker (less precise) than what a normal distribution would give: at k=2, Chebyshev guarantees only 75%, while the normal-distribution rule promises about 95% — the gap reflects the price paid for a universal guarantee rather than one specific to a single distribution shape. Chebyshev's inequality is particularly useful when the real shape of a distribution is unknown or clearly non-normal (highly skewed data, the presence of outliers), a situation where applying a normal-distribution rule directly would give a misleading estimate — a risk similar to the one flagged on our kurtosis coefficient calculator, which also helps reveal when a distribution's shape departs from normality.
Example: k = 2 standard deviations
Inputs
Number of standard deviations (k): 2.
Calculation
Minimum proportion = (1 − 1/2²) × 100 = (1 − 0.25) × 100 = 75%.
Result
Regardless of the distribution, at least 75% of the data lies within 2 standard deviations of the mean.
Frequently asked questions
Why does Chebyshev give a weaker guarantee than the normal-distribution rule?
Because Chebyshev makes no assumption about the shape of the distribution, while a rule like the 68-95-99 one specifically assumes a normal (bell-shaped) distribution. A guarantee valid for absolutely every possible distribution must necessarily be more conservative (and therefore weaker) than a guarantee computed for one specific distribution shape, where much more specific mathematical properties can be exploited.
Is this inequality exact, or only a minimum bound?
It is a minimum bound, not an exact value: for the vast majority of real-world distributions, the actual proportion of data within the interval is well above what Chebyshev guarantees at minimum. The inequality nonetheless remains mathematically true for absolutely any distribution with a finite mean and standard deviation, which makes it a theoretical safety net rather than a precise estimate.
Why must k be greater than 1 for the result to be useful?
At k=1, the formula gives 1 − 1/1² = 0%, a completely trivial and uninteresting guarantee (it says no more than "at least 0% of the data is close to the mean"). The inequality only becomes informative once k is strictly greater than 1, where the guaranteed proportion starts to meaningfully exceed 0%.