Chebyshev's Theorem Formula and Worked Examples

Chebyshev’s theorem says that for any data set or probability distribution with a finite mean and a finite, nonzero standard deviation, at least 1 − 1/k² of the values lie within k standard deviations of the mean, for any k greater than 1. It needs no assumption about the shape of the distribution, which is what separates it from the normal-distribution rules of thumb.

What is Chebyshev’s theorem?

Chebyshev’s theorem (also called Chebyshev’s inequality) is a guaranteed minimum for how much of a distribution sits near its mean. Two numbers are enough to apply it: the mean and the standard deviation. For any data set, at least 75% of the data is within two standard deviations of the mean, and at least 89% is within three (OpenStax, Introductory Statistics 2e, section 2.7). The 89% is a rounded 88.89%, as the arithmetic below shows.

The word “at least” matters. The theorem is a floor, not an estimate. If you know the data are roughly bell-shaped, the empirical rule gives much tighter figures; Chebyshev is the tool for skewed, multimodal or unknown shapes.

Chebyshev’s theorem formula

For a random variable X with finite mean μ and finite nonzero variance σ², and any real k > 0:

Pr(|X − μ| ≥ kσ) ≤ 1/k²

Taking the complement gives the form most textbooks use for data:

Proportion within k standard deviations of the mean ≥ 1 − 1/k²

What each symbol means

X: the variable (a measurement, a score, a count).
μ: the mean. For sample data, use the sample mean.
σ: the standard deviation. For sample data, use the sample standard deviation s.
k: how many standard deviations from the mean you are looking at. It does not have to be a whole number.
1/k²: the largest share that can fall k or more standard deviations away.

Range form

If you start from a distance a rather than a multiple of σ, the same statement reads Pr(|X − μ| ≥ a) ≤ σ²/a². Setting a = kσ turns one form into the other.

Why only k greater than 1 is useful

At k = 1 the bound is 1 − 1/1 = 0, and for k below 1 it is negative. The statement is still true, but it only says “at least 0% or less”, which tells you nothing. Use k > 1.

Chebyshev’s theorem examples

Example 1: a band around the mean

Exam scores have a mean of 70 and a standard deviation of 8, with no assumption about their shape. What share of scores must lie between 54 and 86?

Distance from the mean to each end: 70 − 54 = 16 and 86 − 70 = 16.
k = 16 / 8 = 2.
Minimum share = 1 − 1/2² = 1 − 1/4 = 0.75.

At least 75% of the scores fall between 54 and 86. At most 25% fall outside it.

Example 2: solving for k from a range

With the same mean of 70 and standard deviation of 8, what share must lie between 50 and 90?

a = 90 − 70 = 20, so k = 20 / 8 = 2.5.
Minimum share = 1 − 1/2.5² = 1 − 1/6.25 = 1 − 0.16 = 0.84.

At least 84% of the scores fall between 50 and 90.

Example 3: checking a real sample

Take 15 values: 2, 3, 3, 4, 4, 4, 5, 5, 6, 7, 8, 9, 10, 12, 25. The data are right-skewed, so the normal-distribution rules do not apply.

Sum = 107, so the mean = 107 / 15 = 7.13.
Sum of squared values = 1,219. Sum of squared deviations = 1,219 − 107²/15 = 1,219 − 763.27 = 455.73.
Sample variance = 455.73 / 14 = 32.55, so s = 5.71.
Two standard deviations: 7.13 ± 2(5.71) = 7.13 ± 11.41, an interval from −4.28 to 18.54. Fourteen of the 15 values (93.3%) lie inside; only 25 lies outside.
Chebyshev guarantees at least 75%, and the data comfortably exceed it.

At k = 3 the interval is 7.13 ± 17.12, from −9.98 to 24.25. The value 25 still falls outside, so 14 of 15 (93.3%) are inside against a guaranteed 88.89%. The guarantee held, but loosely. That gap is normal: the bound has to work for every possible distribution, so for any particular one it is usually conservative.

Can the bound ever be exact?

Yes, which is why 1/k² cannot be improved without more assumptions. Let X equal −1, 0 or +1 with probabilities 1/(2k²), 1 − 1/k² and 1/(2k²). For k = 2 these are 1/8, 3/4 and 1/8. The mean is 0 and the variance is 2 × (1/8) = 0.25, so σ = 0.5. The chance that X is 2σ = 1 or more away from the mean is 1/8 + 1/8 = 1/4 = 1/k². Equality is reached.

How to interpret the result

Read the result as “at least this much is inside the band”. The table compares the Chebyshev floor with the empirical rule, which only applies to bell-shaped, symmetric data.

Distance from mean (k)	Chebyshev: at least within	Chebyshev: at most outside	Empirical rule (bell-shaped data)
1.5	55.56%	44.44%	not stated for 1.5
2	75%	25%	approx. 95%
2.5	84%	16%	not stated for 2.5
3	88.89%	11.11%	more than 99%
4	93.75%	6.25%	not stated for 4
5	96%	4%	not stated for 5

The Chebyshev columns come from 1 − 1/k² and 1/k². The empirical-rule figures are those given in OpenStax section 2.7. If the data do turn out to be bell-shaped, the empirical rule is the better description; Chebyshev remains a valid but looser statement.

Assumptions and common mistakes

Finite mean and variance. The theorem requires a finite mean and a finite, nonzero variance. For a real sample these are always finite, but the figure then describes the sample, not an unseen population.
Treating “at least” as “about”. 75% at k = 2 is a minimum. Do not report it as the expected share.
Using k of 1 or less. The bound is 0 or negative, so it says nothing.
Applying it to one tail. The bound covers both tails together. It does not say how the 1/k² splits between them.
Mixing population and sample standard deviation. Compute σ or s consistently and say which you used. Example 3 uses the sample standard deviation s (divisor n − 1).
Forgetting outliers. A single extreme value inflates the standard deviation and widens every Chebyshev interval. See winsorizing vs. trimming for handling them before summarising.

How to report it (APA example)

Chebyshev’s theorem is usually used to justify a statement rather than reported as a result of its own. Report the descriptive statistics first, then the guarantee. APA style sets statistical symbols such as M and SD in italics. A sentence for Example 3 could read:

Scores were right-skewed (M = 7.13, SD = 5.71, N = 15). Because normality could not be assumed, Chebyshev’s theorem was used: at least 75% of scores were guaranteed to fall within two standard deviations of the mean, and 93.3% did.

For context on how these summary statistics are chosen and reported, see the guide to descriptive versus inferential statistics.

How to calculate it in Excel, R or Python

Excel

With the data in A2:A16 and k in D1:

Mean:           =AVERAGE(A2:A16)
Sample SD:      =STDEV.S(A2:A16)
Minimum share:  =1-1/D1^2
Lower / upper:  =AVERAGE(A2:A16)-D1*STDEV.S(A2:A16)   and   =AVERAGE(A2:A16)+D1*STDEV.S(A2:A16)
Actual share:   =COUNTIFS(A2:A16,">="&F1,A2:A16,"<="&F2)/COUNT(A2:A16)

Here F1 and F2 hold the lower and upper limits.

R

x <- c(2,3,3,4,4,4,5,5,6,7,8,9,10,12,25)
k <- 2
bound <- 1 - 1/k^2
lo <- mean(x) - k*sd(x); hi <- mean(x) + k*sd(x)
actual <- mean(x >= lo & x <= hi)
c(bound = bound, actual = actual)

In R, sd() is the sample standard deviation (divisor n − 1).

Python

import statistics as st
x = [2,3,3,4,4,4,5,5,6,7,8,9,10,12,25]
k = 2
bound = 1 - 1/k**2
m, s = st.mean(x), st.stdev(x)      # stdev = sample SD
lo, hi = m - k*s, m + k*s
actual = sum(lo <= v <= hi for v in x) / len(x)
print(bound, round(actual, 3))      # 0.75 0.933

Frequently asked questions

What is Chebyshev’s theorem in simple terms?

It guarantees a minimum share of data near the mean for any distribution: at least 1 − 1/k² of values fall within k standard deviations, so at least 75% within 2 and at least 88.89% within 3.

What is the formula for Chebyshev’s theorem?

Proportion within k standard deviations ≥ 1 − 1/k², equivalently Pr(|X − μ| ≥ kσ) ≤ 1/k², for k > 1.

When should I use Chebyshev’s theorem instead of the empirical rule?

Use Chebyshev when the data are skewed, have several peaks, or you do not know the shape. Use the empirical rule only when the data are approximately bell-shaped and symmetric.

Does k have to be a whole number?

No. Any real k greater than 1 works, as in the k = 2.5 example above.

Is it the same as a confidence interval?

No. It describes where individual values fall, not where a population parameter lies. See confidence interval for the latter.

Sources

OpenStax. Introductory Statistics 2e, section 2.7, “Measures of the Spread of the Data.” openstax.org/books/introductory-statistics-2e/pages/2-7-measures-of-the-spread-of-the-data (checked 2026-10-29). Source for the 75% and 89% statements and the empirical-rule percentages.
“Chebyshev’s inequality,” Wikipedia. en.wikipedia.org/wiki/Chebyshev’s_inequality (checked 2026-10-29). Reference article for the formal statement, the finite-variance condition, and the equality example.

Chebyshev’s Theorem Formula: 1 – 1/k² With Examples