Chebyshev’s Theorem Calculator (minimum share within k standard deviations)

Understanding the Chebyshev's theorem calculator — concept and principle

This calculator applies Chebyshev's theorem: for any data set or distribution with a finite standard deviation, at least 1 − 1/k² of the values lie within k standard deviations of the mean. Enter k, or the share you want covered, plus the mean and standard deviation, and it gives the guaranteed minimum share, the interval and a comparison with the normal distribution.

How it is calculated

Minimum share within k SD = 1 − 1/k² (for k > 1)

Maximum share outside = 1/k²

Interval = mean ± k × SD

Working backwards: k = 1 ÷ √(1 − share)

• For a normal distribution the share within k SD is erf(k ÷ √2), shown for comparison.

The principle

Chebyshev's inequality comes from a simple counting argument. The variance is the average squared distance from the mean. If more than 1/k² of the values were at least k standard deviations away, those values alone would contribute more than (1/k²) × (kσ)² = σ² to the average squared distance, which is more than the whole variance. So no distribution can have more than 1/k² of its values that far out. The bound works for any shape, which is why it is usually far from tight.

Worked example

Default inputs: k = 2, mean 70, SD 10.

1. Minimum share within 2 SD = 1 − 1/2² = 75 %.
2. Maximum share outside = 1/4 = 25 %; k used 2.
3. Interval = 70 ± 2 × 10 = 50 to 90, width 40.
4. For a normal distribution about 95.45 % would lie within 2 SD.

How to read the result

The Chebyshev figure is a guarantee, not an estimate. If you know nothing about the shape of your data, you can still say at least 75 % lie within 2 SD and at least 88.9 % within 3 SD. If the data are roughly bell-shaped, the real share is much higher, closer to the normal comparison. For k ≤ 1 the bound is 0 and tells you nothing.

Common mistakes

• Treating 75 % as the expected share instead of a minimum.
• Using it with k ≤ 1.
• Using the empirical 68–95–99.7 rule for data that are not bell-shaped.
• Confusing the population SD with a rough guess.

Frequently asked questions

What does Chebyshev's theorem say?

At least 1 − 1/k² of any distribution lies within k standard deviations of the mean.

What share is within 3 standard deviations?

At least 88.9 % for any distribution; about 99.7 % for a normal one.

How is it different from the empirical rule?

The empirical rule applies to bell-shaped data; Chebyshev applies to any data but gives weaker bounds.

Sources: Mathematics LibreTexts, Interactive Lecture Notes for Introductory Statistics, §3.4 Interpreting the Mean and Standard Deviation