Chebyshev's Theorem Calculator

What this Chebyshev's theorem calculator does

Chebyshev's theorem, also called Chebyshev's inequality, puts a guaranteed floor under the share of values that sit close to the mean. It needs only the mean and the standard deviation, so it works for skewed, lumpy or unknown distributions where the normal curve would be a poor assumption.

The calculator has three modes. Enter k to get the minimum coverage and the maximum share outside. Enter a target percentage to get the k that guarantees it. Or enter a mean, a standard deviation and the two ends of an interval to see what the theorem guarantees for that interval. Every result lists the substituted formula so you can check it by hand. For the bell-shaped case, see the Normal Distribution Calculator and the Empirical Rule Calculator.

Inputs and what each one means

Pick the mode that matches what you know. The mean and standard deviation are optional in the first two modes and required in the third.

k

How many standard deviations from the mean the interval reaches on each side. It can be any positive number, including a decimal.

Mean

The average of the data, or the expected value of the distribution. Optional in the first two modes, where it only turns k into interval bounds.

Standard deviation (SD)

The usual measure of spread, in the same units as the mean. It must not be negative.

Target minimum coverage

The share of values you want guaranteed to fall inside the interval, entered as a percentage from 0 up to but not including 100.

Lower and upper endpoint

The ends of an interval around the mean, used only in the interval mode. The mean must lie strictly between them.

Keep the mean, the standard deviation and any endpoints in the same units. The standard deviation can come from a population or from a sample, and this page does not compute it from raw data; the Standard Deviation Calculator is the place for that.

The Chebyshev's theorem formula

The theorem is usually stated as a bound on the tails: P(|X − μ| ≥ kσ) ≤ 1/k². Turning that around gives the share inside the interval, which is the form the calculator reports.

• Minimum proportion within k SD = 1 − 1/k² (for k > 1; otherwise 0)
• Minimum percentage = (1 − 1/k²) × 100%
• Maximum proportion at least k SD away = 1/k², both tails combined
• Interval = mean − k × SD to mean + k × SD
• Required k for a target proportion p = 1 / √(1 − p), for 0 ≤ p < 1
• k from an interval = distance from the mean to an endpoint / SD

The bound counts both tails together. It says that no more than 1/k² of the values can be k or more standard deviations from the mean, but it does not say how that share is split between the low and high side. The reverse formula comes from solving 1 − 1/k² = p for k, which is why a target of 100% is impossible: no finite k makes 1/k² equal zero.

Step-by-step: how the result is calculated

In the first mode the calculator squares k, takes its reciprocal, subtracts that from 1 and converts the result to a percentage. It then reports the complement, 1/k², as the most that can fall outside. If a mean and standard deviation are given, it adds k × SD to the mean and subtracts k × SD from it to get the interval bounds. The decimal places menu changes only what is displayed, never the arithmetic.

Worked example: k = 2

Suppose a set of scores has a mean of 50 and a standard deviation of 8 (illustrative figures). With k = 2, the minimum proportion is 1 − 1/4 = 0.75, so at least 75% of the scores lie between 50 − 2 × 8 = 34 and 50 + 2 × 8 = 66. At most 25% lie outside that range.

Worked example: a decimal k

With the same mean and standard deviation and k = 2.5, k² = 6.25 and 1/6.25 = 0.16, so at least 84% of the values lie between 30 and 70, and at most 16% lie outside.

Worked example: working backwards from a target

To guarantee at least 90%, k = 1 / √(1 − 0.9) = 1 / √0.1 ≈ 3.1623. With a mean of 50 and a standard deviation of 8 the interval is 50 ± 3.1623 × 8, which is about 24.70 to 75.30. That is the distance the theorem needs, not the exact range holding 90% of the data. For real data that range is usually narrower.

Common values of k

These values hold for any distribution with a finite mean and standard deviation. The last column shows the empirical rule for comparison; it applies only to roughly bell-shaped data.

Reading the result

The headline figure is a minimum, and the calculator labels it that way. The share outside is a maximum. The meter under the results shows the guaranteed share, and anything not shaded might still be inside, because the real distribution is rarely the worst case. A few points to keep in mind when you read a result:

• It is a guarantee, not an estimate. The true share is at least the figure shown and is often well above it.
• Both tails are combined. The theorem does not say how much of the outside share lies below or above the mean.
• The interval is symmetric. It always runs the same number of standard deviations on each side of the mean.
• Interval bounds are in the original units. They are the mean minus k × SD and the mean plus k × SD.

Entering an interval instead of k

If you know the interval, the interval mode finds k for you by dividing the distance from the mean to an endpoint by the standard deviation. With a mean of 60, a standard deviation of 2 and an interval from 56 to 64, each endpoint is 4 units, or 2 standard deviations, from the mean, so at least 75% of the values lie in that interval.

The theorem describes intervals centred on the mean, so an uncentred interval is not treated as if it were symmetric. Suppose the mean is 50, the standard deviation is 5 and the interval is 35 to 70. The endpoints are 3 and 4 standard deviations from the mean. The calculator uses the smaller distance, k = 3, which gives the symmetric interval 35 to 65. That interval sits inside yours, so its guarantee of at least 88.89% holds for yours as well, though the true share in the wider interval is likely higher.

Edge cases and input checks

The calculator stops and explains the problem, instead of showing a misleading number, when:

• a field is empty where a value is required, or does not contain a number;
• k is zero or negative, or larger than 1,000,000;
• the standard deviation is negative;
• the target coverage is negative or is 100% or more;
• an interval's endpoints are in the wrong order, or the mean is not strictly between them.

Two cases are handled rather than rejected. When k is 1 or less, the guaranteed minimum is 0%, which is reported with a note that the theorem says nothing useful there. When the standard deviation is zero in the first two modes, every value equals the mean and the interval collapses to that single point; the result says so.

Chebyshev's theorem versus the empirical rule

The empirical rule says that about 68%, 95% and 99.7% of values lie within one, two and three standard deviations of the mean, but only for data that follow a roughly normal curve. Chebyshev's theorem needs no such assumption, so its figures are lower: at least 75% and 88.89% at two and three standard deviations. Use the empirical rule when a bell shape is reasonable and Chebyshev's theorem when the shape is unknown, skewed or doubtful, or when you need a statement that cannot fail. For checking how far a single value sits from the mean, the Z-score Calculator gives the same distance in standard deviations.

Assumptions and limitations

• The distribution or data set must have a finite mean and standard deviation. The theorem does not apply to distributions without a finite variance.
• The bound is conservative. Many real data sets have far more values near the mean than the minimum suggests.
• For a sample, the result describes the values in that sample, based on its own mean and standard deviation. It is not a guarantee about a wider population.
• The bound covers both tails together. Tighter one-sided bounds exist, but this calculator does not compute them.
• The result is only as good as the standard deviation you enter. An estimated or rounded value passes its error on to the interval.
• This calculator provides a mathematical result from the values you enter. It is an informational and educational tool, not professional or statistical advice. For decisions that matter, consult a qualified professional.

Formula sources

The theorem, its reverse form and the sample version were checked against these independent references:

• Wolfram MathWorld, Chebyshev Inequality
• University of California, Riverside, MATH 149A notes, Chebyshev's inequality (the theorem as stated in Hogg, McKean and Craig, Introduction to Mathematical Statistics)
• College of William & Mary, MATH 451/551, Chapter 3.5: Inequalities
• Emory University, MATH 117, Chebyshev's Theorem, including the version for a sample

Frequently asked questions