Universal probability bounds for any distribution with a defined mean and variance.
Range Parameters
Mathematical Formula
P(|X - μ| < kσ) ≥ 1 - 1/k²
Guarantees the lower bound of values within k standard deviations.
Probability Bounds
Minimum Guaranteed75.0%Theoretical Lower Bound
Normal Distribution95.4%Empirical Rule (Bell Curve)
Maximum Outliers25.0%Allowed Outside Range
Observation
No matter how skewed your data is, at least 75.0% of it MUST be within 2 standard deviations.
About this calculator
Overview
Unlike the Empirical Rule (68-95-99.7) which only works for Normal distributions, Chebyshev's Inequality is a universal mathematical guarantee. It provides the absolute minimum percentage of points that must lie within k standard deviations for ANY distribution.
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Pro Tips
This rule only provides useful bounds when k > 1.0.
For many distributions, the actual percentage is much higher than this lower bound.
It works even if the distribution has multiple peaks or is highly asymmetric.
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Fun Facts
"Auditors use this when they encounter 'messy' data that doesn't look like a bell curve."
"For k=2, it guarantees that at least 75% of values are within 2 standard deviations, no matter how skewed the data is."
"Named after Pafnuty Chebyshev, a Russian mathematician who founded many analytical theories for risk algorithms."
