Chebyshev's Theorem Calculator — Proportion Within k Standard Deviations
Chebyshev's theorem guarantees that for any distribution (not just normal), at least 1 − 1/k² of the data falls within k standard deviations of the mean. Enter k (must be > 1), a mean, and standard deviation to get the minimum proportion, the exact interval, and a visualisation.
At least 75% of data lies within 2σ of the mean (any distribution)
Apply the formula
Substitute k
Minimum percentage within interval
Interval [μ − kσ, μ + kσ]
- 1
k² (square of k)
2² = 4 - 2
1 ÷ k²
1 ÷ 4 = 0.25 - 3
Minimum proportion inside (1 − 1/k²)
1 − 0.25 = 0.75 - 4
Minimum percentage inside
0.75 × 100 = 75At least this percentage of any distribution lies within k standard deviations of the mean.
How does this calculator work?
Chebyshev's inequality: at least (1 − 1/k²) × 100% of any distribution's data lies within k standard deviations of the mean (k > 1). For k = 2: ≥75%; k = 3: ≥88.9%; k = 4: ≥93.75%. Unlike the 68-95-99.7 rule, this bound is valid for every distribution with finite mean and variance, not just the normal distribution.
Formula
How this is calculated
Chebyshev's inequality is a universal probabilistic bound that holds for any probability distribution with a finite mean μ and finite variance σ². Unlike the empirical (68-95-99.7) rule which is specific to normal distributions, Chebyshev's theorem makes no assumption about the shape of the distribution. It states: the probability that a randomly selected value lies more than k standard deviations from the mean is at most 1/k². Equivalently, at least 1 − 1/k² of the distribution is within k standard deviations.
For k = 2 the bound says at least 75% of data lies within 2σ of the mean; for k = 3 it is at least 88.9%; for k = 4 it is at least 93.75%. These guarantees are conservative — for a normal distribution the corresponding percentages are 95.4%, 99.7%, and 99.99%. Chebyshev's bound is tight: there exist distributions for which the bound is achieved exactly.
The theorem is most useful when the distribution is unknown or non-normal. In practice it appears in outlier detection (any value more than k standard deviations from the mean is guaranteed to represent at most 1/k² of the population), in sample-size estimation, and in proving the Law of Large Numbers. The result requires only that the mean and variance exist and are finite.
Frequently asked questions
At k = 1 the bound gives 1 − 1/1 = 0, which is trivially true but useless. For k < 1 the formula gives a negative bound, which is meaningless. The bound only becomes informative (a non-trivial positive lower bound on the proportion) when k > 1.
The empirical (68-95-99.7) rule applies only to perfectly normal distributions and gives exact percentages. Chebyshev's theorem applies to any distribution but gives weaker (more conservative) guarantees: at least 75% within 2σ (vs. 95.4% for normal), at least 88.9% within 3σ (vs. 99.7%). Use the empirical rule when you know the data is normally distributed; use Chebyshev's when you don't.
Any distribution with a defined (finite) mean and finite variance — which covers virtually all real-world data sets. Distributions without a finite variance (e.g. Cauchy) are excluded. In practice, if you can compute a mean and a standard deviation from your data, Chebyshev's inequality applies to the empirical distribution.
TG we-Calculate Editorial Team. (2026). Chebyshev's Theorem Calculator — Proportion Within k Standard Deviations [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/chebyshevs-theorem-calculator
TG we-Calculate Editorial Team. "Chebyshev's Theorem Calculator — Proportion Within k Standard Deviations." TG we-Calculate. 2026. https://we-calculate.com/calculator/chebyshevs-theorem-calculator.
TG we-Calculate Editorial Team, "Chebyshev's Theorem Calculator — Proportion Within k Standard Deviations," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/chebyshevs-theorem-calculator
@misc{wecalculate_chebyshevs_theorem_calculator, title = {Chebyshev's Theorem Calculator — Proportion Within k Standard Deviations}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/chebyshevs-theorem-calculator}}, year = {2026}, note = {TG we-Calculate} }
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