Chebyshev's Theorem Calculator
Estimate guaranteed coverage using Chebyshev’s theorem for any distribution.
Table of Contents
How to Use
- Enter the population mean and standard deviation that describe your dataset or process.
- Specify the k value (number of standard deviations from the mean) and the sample size you plan to analyze.
- Click calculate to view the interval bounds, guaranteed coverage percentage, and maximum number of potential outliers.
About Chebyshev's Theorem
Chebyshev’s theorem provides a distribution-free guarantee about how much of the data lies within k standard deviations of the mean. Unlike the empirical rule, it does not assume normality.
- Works for any distribution with finite variance.
- Requires k > 1 (one standard deviation is not guaranteed).
- Guarantees at least 1 − 1/k² of observations within k standard deviations.
Practical Uses
Use Chebyshev’s theorem when the distribution is unknown or highly skewed. It offers conservative bounds for quality control, risk assessment, and validating minimum coverage requirements.
Frequently Asked Questions
- Why must k be greater than 1?
- Chebyshev’s inequality provides meaningful guarantees only for k > 1. At exactly one standard deviation, the bound becomes zero and offers no information.
- How does this compare to the empirical rule?
- The empirical rule (68-95-99.7 rule) assumes a normal distribution. Chebyshev’s bound is weaker but applies to any distribution, making it safer when the shape is unknown.
- What if my sample size is small?
- Chebyshev’s theorem still holds, but the minimum counts may be low. Collect more data to tighten the guaranteed coverage or pair the result with additional distribution knowledge.
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