Chebyshev’s Rule Calculator

Instructions: This Chebyshev's Rule calculator will show you how to use Chebyshev's Inequality to estimate probabilities of an arbitrary distribution. You can estimate the probability that a random variable \(X\) is within \(k\) standard deviations of the mean, by typing the value of \(k\) in the form below; OR specify the population mean \(\mu\), population standard deviation \(\sigma\) and the even \((a,b)\) for which you want to estimate the probability:

Type of the value of k (number of standard deviations from the mean)

Population Mean (\(\mu\))
Population St. Dev. (\(\sigma\))
Lower Limit of the event \((a)\):
Upper Limit of the event \((b)\):

More About the Chebyshev's Inequality Calculator

We use Chebyshev's inequality to compute the probability that \(X\) is within \(k\) standard deviations of the mean. According to Chebyshev's rule, the probability that \(X\) is within \(k\) standard deviations of the mean can be estimated as follows:

\[ \Pr(|X - \mu| < k \sigma) \ge 1 - \frac{1}{k^2} \]

Chebyshev's inequality is very powerful, because it applies to any generic distribution. If you are dealing with a normal distribution, you should use our empirical rule instead .

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