Chebyshev's Theorem

( ≥ ) ≤

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Calculate

Example: salaries at a company have a mean of $3,000 and a standard deviation of $500, but the shape of the distribution is unknown. At minimum, what share of salaries falls between $2,000 and $4,000? Change the values to work out your own case.

Mean μ Standard deviation σ Standard deviations k

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What each term means

Probability (P): The probability of an event. In Chebyshev's theorem, it is the probability that the random variable lies a certain distance from the mean.
Absolute Value |X - μ|: The distance between the random variable X and its mean μ. The absolute value measures the distance regardless of direction (above or below the mean).
Constant k (k): A positive real number (k > 0) giving how many standard deviations we are considering. For example, k = 2 means 2 standard deviations from the mean.
Standard Deviation (σ): The standard deviation of the random variable X, which measures how spread out the data are around the mean. It is the square root of the variance.
One (1): The numerator of the fraction that bounds the maximum probability.
k Squared (k²): The denominator of the fraction. The larger k is, the smaller 1/k² gets, meaning values far from the mean are less likely. For example, with k = 2, P(|X - μ| ≥ 2σ) ≤ 1/4 = 0.25: at most 25% of the data lie more than 2 standard deviations from the mean.