Chebyshev Probability Bound
Bound the share of a distribution within k standard deviations.
Your numbers
At least within k standard deviations
Requires a finite mean and variance. This is a universal bound, not the exact probability of a particular distribution.
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Formula
P(|X−μ|<kσ)≥1−1/k²
Bound the share of a distribution within k standard deviations.
Worked example
- Standard deviations k
- 2
- At least within k standard deviations
- 75 %
- At most outside the interval
- 25 %
Scope and limitations
Requires a finite mean and variance. This is a universal bound, not the exact probability of a particular distribution.
Inputs are checked for missing values and allowed ranges. A result outside the finite numeric range is not displayed. Rounding affects display, while the calculation uses the model’s numeric precision.
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Formula version: b8b3271de52a · Example checked: