Chebyshev's Theorem Calculator
Calculate minimum data percentage within k standard deviations using Chebyshev's inequality. Works for any distribution shape.
About
Chebyshev's inequality provides a guaranteed lower bound on data concentration around the mean for any distribution - normal, skewed, bimodal, or otherwise. The bound states that at least 1 − 1/k2 of observations fall within k standard deviations (σ) of the mean (μ), provided k > 1. This is weaker than distribution-specific rules (e.g., the empirical 68-95-99.7 rule for normal curves) but applies universally. Misapplying the empirical rule to non-normal data leads to incorrect confidence intervals and flawed risk assessments.
This calculator accepts a mean, standard deviation, and a k value to compute the minimum guaranteed percentage of data within the resulting interval. It also computes the explicit numeric range [μ − kσ, μ + kσ]. Note: the theorem provides a lower bound only. Actual concentration may be significantly higher depending on the true distribution shape. The bound is not tight for most practical distributions.
Formulas
Chebyshev's inequality (also Chebyshev-Bienaymé inequality) bounds the probability that a random variable deviates from its mean by more than k standard deviations:
Equivalently, the minimum proportion of data within the interval is:
The numeric interval boundaries are computed as:
Where μ = population or sample mean, σ = standard deviation (population or sample), k = number of standard deviations (k > 1 for a non-trivial bound). The theorem requires only finite mean and finite non-zero variance. No assumption about distribution shape is needed.
Reference Data
| k (Std Devs) | Min. % Within Range | Max. % Outside | Normal Dist. Actual % | Bound Tightness |
|---|---|---|---|---|
| 1.0 | 0.00% | 100.00% | 68.27% | Non-informative |
| 1.1 | 17.36% | 82.64% | 72.87% | Very loose |
| 1.2 | 30.56% | 69.44% | 76.99% | Loose |
| 1.3 | 40.83% | 59.17% | 80.64% | Loose |
| 1.4 | 48.98% | 51.02% | 83.85% | Moderate |
| 1.5 | 55.56% | 44.44% | 86.64% | Moderate |
| 2.0 | 75.00% | 25.00% | 95.45% | Moderate |
| 2.5 | 84.00% | 16.00% | 98.76% | Moderate |
| 3.0 | 88.89% | 11.11% | 99.73% | Tighter |
| 3.5 | 91.84% | 8.16% | 99.95% | Tighter |
| 4.0 | 93.75% | 6.25% | 99.99% | Tight |
| 4.5 | 95.06% | 4.94% | ≈ 100% | Tight |
| 5.0 | 96.00% | 4.00% | ≈ 100% | Tight |
| 6.0 | 97.22% | 2.78% | ≈ 100% | Very tight |
| 7.0 | 97.96% | 2.04% | ≈ 100% | Very tight |
| 8.0 | 98.44% | 1.56% | ≈ 100% | Very tight |
| 10.0 | 99.00% | 1.00% | ≈ 100% | Near-maximum |