Distribution Variance Calculator

About this calculator

For a discrete random variable X with possible values x_i and probabilities p_i, the expected value (mean) is μ = Σ x_i·p_i, and the variance is Var(X) = Σ p_i·(x_i − μ)² — equivalently E[X²] − μ² using the shortcut formula. The standard deviation is simply the square root of the variance, expressed in the same units as X.

These three numbers describe both the center and the spread of a distribution: variance and standard deviation quantify how much outcomes typically deviate from the expected value, which is central to risk analysis, insurance pricing, quality control, and any statistics or probability coursework covering random variables.

Enter each value alongside its probability (the probabilities should sum to 1) and this calculator computes the mean, variance, and standard deviation directly from the definition.