Table of Contents
What Is Chebyshev's Theorem?
Chebyshev's inequality (also Chebyshev's theorem) states that for ANY distribution with a finite mean and variance, at least 1-1/k² of the data falls within k standard deviations of the mean, for any k > 1. Unlike the empirical rule (68-95-99.7), which only applies to normal distributions, Chebyshev's works universally.
This makes it invaluable when the distribution shape is unknown or non-normal. The bounds are conservative -- the actual percentage within kσ is usually higher than the Chebyshev minimum.
Formula
Common Values
| k | Chebyshev Min | Normal (actual) |
|---|---|---|
| 1.5 | 55.56% | 86.64% |
| 2 | 75.00% | 95.45% |
| 3 | 88.89% | 99.73% |
| 4 | 93.75% | 99.994% |
FAQ
When should I use Chebyshev vs the empirical rule?
Use the empirical rule (68-95-99.7) when data is normally distributed. Use Chebyshev when the distribution is unknown, skewed, or non-normal. Chebyshev gives a guaranteed minimum; the empirical rule gives approximate values for normal data only.
Why must k be greater than 1?
At k=1, the formula gives 1-1/1=0%, which is trivially true but unhelpful. For k<1, the formula would give a negative proportion, which is meaningless.