# Chebyshev’s Theorem Calculator

URL: https://speedycalculator.com/calculators/statistics/chebyshevs-theorem/
Last updated: 2026-10-30

> The Chebyshev’s theorem calculator gives the guaranteed minimum share of any distribution within k standard deviations of the mean, and the maximum probability of a value lying further away – from k directly, or from a distance and the variance.

## Inputs

- Input
- Mean (μ)
- Number of standard deviations (k)
- Standard deviation (σ)
- Distance from the mean (d)
- Variance (σ²)

## Results

- At least this share lies within the interval
- k (distance in standard deviations)
- At most this share lies outside (|X − μ| ≥ kσ)
- Interval lower end
- Interval upper end
- Share within the interval if the data were normal

## Formula

```
P(|X − μ| ≥ kσ) ≤ 1 ÷ k²  ·  P(|X − μ| < kσ) ≥ 1 − 1 ÷ k²
In terms of a distance d: P(|X − μ| ≥ d) ≤ σ² ÷ d²
```

## Explanation

Chebyshev’s inequality is a safety net for situations where you know the mean and standard deviation of a variable but nothing about the shape of its distribution. Whatever the shape – skewed, bimodal or heavy-tailed – at least 1 − 1/k² of the values lie within k standard deviations of the mean. So at least 75% lie within 2σ, at least 88.9% within 3σ and at least 93.75% within 4σ.

You can enter the number of standard deviations k directly together with the mean and standard deviation, or enter a distance from the mean and the variance (the form P(|X − μ| ≥ d) ≤ σ²/d²). The calculator returns the upper bound on the probability of being that far out, the lower bound on the probability of being inside, the interval itself, and – for comparison – what a normal distribution would give.

The bound is deliberately conservative. For a normal distribution 95.45% lies within 2σ, well above Chebyshev’s 75%. For k ≤ 1 the theorem says nothing useful, since the bound on the tail probability is 100% or more.

### How to use the Chebyshev’s Theorem Calculator

- **Input**
- **Mean (μ)**
- **Number of standard deviations (k)**
- **Standard deviation (σ)**
- **Distance from the mean (d)**
- **Variance (σ²)**

Results update as you type, and everything is calculated in your browser.

### How it works

Convert the distance into standard deviations, k = d ÷ σ, then the tail bound is 1/k² (capped at 100%) and the coverage bound is 1 − 1/k² (at least 0). The interval is mean ± k standard deviations.

### Worked example

Exam scores have mean 50 and standard deviation 10, shape unknown. With k = 2 the interval is 50 ± 20 = 30 to 70, and at least 1 − 1/2² = 75% of students scored inside it; at most 25% scored outside. If the scores were normal the share would be 95.45%.

Distance form: with variance 100 (σ = 10) and distance 15, k = 1.5, so at most 100/15² = 44.44% of values are 15 or more from the mean and at least 55.56% are within 35 to 65.

### Chebyshev vs normal coverage

kChebyshev: at leastNormal distribution1.555.56%86.64%275.00%95.45%2.584.00%98.76%388.89%99.73%493.75%99.994%

### Inputs explained

FieldWhat to enterInputChoose one: Number of standard deviations k, Distance from the mean and varianceMean (μ)A numberNumber of standard deviations (k)A numberStandard deviation (σ)A numberDistance from the mean (d)A numberVariance (σ²)A number

### What each result means

- **At least this share lies within the interval** = max(0, 1 - 1÷[k]^2)×100
- **k (distance in standard deviations)** = [k]
- **At most this share lies outside (|X − μ| ≥ kσ)** = min(1, 1÷[k (distance in standard deviations)]^2)×100
- **Share within the interval if the data were normal** = (1 - 2×(1 - ncdf([k (distance in standard deviations)])))×100

## Worked example

Inputs — Input: Number of standard deviations k; Mean (μ): 50; Number of standard deviations (k): 2; Standard deviation (σ): 10.

Results — **At least this share lies within the interval = 75 %**; k (distance in standard deviations) = 2; At most this share lies outside (|X − μ| ≥ kσ) = 25 %; Interval lower end = 30; Interval upper end = 70; Share within the interval if the data were normal = 95.45 %.

## Key terms

- **Chebyshev’s inequality**: P(|X − μ| ≥ kσ) ≤ 1/k² for any distribution with finite variance.
- **k**: The distance from the mean measured in standard deviations.
- **Empirical rule**: The 68–95–99.7 percentages that apply to normal distributions only.

## FAQ

**What does Chebyshev’s theorem say?**
For any distribution with a finite variance, at least 1 − 1/k² of the values lie within k standard deviations of the mean.

**What percentage lies within 2 standard deviations?**
At least 75% (1 − 1/4). Within 3 standard deviations at least 88.89%, within 1.5 at least 55.56%.

**How is it different from the empirical rule?**
The 68–95–99.7 rule only holds for normal (bell-shaped) data. Chebyshev’s bound holds for every distribution but is much looser.

**Why does the theorem need k > 1?**
For k ≤ 1 the bound 1/k² is 100% or more, which tells you nothing.

**Can I use a distance instead of k?**
Yes – P(|X − μ| ≥ d) ≤ σ²/d². With variance 100 and distance 15, at most 44.44% of values are 15 or more away from the mean.

**Is the bound ever reached exactly?**
Yes. A distribution with mass only at μ and μ ± kσ can hit it, so the theorem cannot be improved without more information.

**Does it work for samples?**
It applies to any data set when you use the data’s own mean and population standard deviation; for sample estimates it is approximate.


## Sources

- [Penn State STAT 414 – Introduction to Probability Theory](https://online.stat.psu.edu/stat414/)
- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/)
- [Wolfram MathWorld – Probability and statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html)