Popoviciu's inequality

Popoviciu's inequality bounds the variance of a random variable supported on [a,b] by (b−a)²/4, with equality for equal probability at the endpoints.

Variance: Variance is the expected squared deviation of a random variable from its mean. The inequality places a sharp upper bound on this measure of spread.
Expected value: The expected value is the probability-weighted average of a random variable's possible outcomes. The variance is computed relative to this central value.
Concentration inequality: A concentration inequality bounds the probability that a random quantity deviates from a central value. Popoviciu's bound supplies a variance estimate for range-based concentration results.
Chebyshev's inequality: Chebyshev's inequality bounds the probability of a deviation from the mean using variance. Unlike Popoviciu's result, it uses variance to bound tail probabilities, not support to bound variance.
RMS–AM inequality: The root-mean-square of nonnegative numbers is at least their arithmetic mean. Applied to distances from the endpoints, it supplies the quadratic step in a proof.
Probability distribution: A probability distribution specifies the probabilities of outcomes of a random process. The inequality applies across distributions supported on the same interval.
Hoeffding's inequality: Hoeffding's inequality bounds deviations of sums of independent bounded random variables. Its range-based setting makes Popoviciu's variance bound a natural companion.
Bhatia–Davis inequality: A bound states that variance is at most (M−μ)(μ−m) for a variable in [m,M] with mean μ. It uses the actual mean for a tighter bound, while Popoviciu's bound uses only interval width.
Bernoulli distribution: A Bernoulli distribution assigns probabilities to the outcomes zero and one. Equal endpoint probabilities give the rescaled distribution attaining equality.
Support (mathematics): The support of a probability distribution is the smallest closed set containing its outcomes with probability one. The interval endpoints determine the width used in the bound.

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Levinson's inequalityRelated: It belongs to the family of refinements and companions to classical convexity inequalities.