How this calculator works
Chebyshev inequality applies to any distribution with finite variance. It bounds the probability of being at least k standard deviations away from the mean by 1/k². The complementary guarantee is for the open interval between the displayed endpoints; including the endpoints cannot reduce coverage. For k below one, the bound gives only the trivial guarantee of zero. A zero standard deviation is excluded from this standardized calculation.
Formula
P(|X - mean| >= k × sd) <= min(1, 1/k²). Guaranteed inside coverage >= max(0, 1 - 1/k²). For target c, k = 1/sqrt(1 - c).
Worked example
For mean 10 and standard deviation 2, k = 2 gives endpoints 6 and 14 and guarantees at least 75% inside. A 96% guarantee needs k = 5, giving endpoints 0 and 20.
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Frequently asked questions
Is this the normal-distribution 68-95-99.7 rule?
No. Normality supports tighter percentages. Chebyshev uses only a finite mean and variance.
Can I enter a sample mean and standard deviation?
You can use them descriptively, but sample estimates do not by themselves establish a population probability guarantee.
Why is 100% coverage rejected in inverse mode?
The formula needs an infinite multiple for a 100% guarantee. Finite variance does not imply bounded support.