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Discrete probability-model mean and variance

For a finite distribution, the mean is the probability-weighted outcome and the variance is the probability-weighted squared deviation from that mean.

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Calculate the weighted mean

Let outcomes xᵢ have nonnegative probabilities pᵢ summing to one. The mean uses those probabilities, rather than giving each listed outcome equal weight by default.

μ=∑ipixi\mu=\sum_i p_i x_i

An equal arithmetic mean is the special case in which every listed outcome has the same probability.

Use the same mean inside the squared deviations

Variance measures dispersion around μ, not around an arbitrary target or a separately quoted forecast.

σ2=∑ipi(xi−μ)2,σ=σ2\sigma^2=\sum_i p_i(x_i-\mu)^2,\qquad\sigma=\sqrt{\sigma^2}

The probability weights already define the distribution; there is no additional n or n − 1 divisor in this formula.

Use two simple boundary checks

If every outcome has the same value, each deviation from the mean is zero and variance is zero. Adding a constant to every outcome changes the mean but leaves variance unchanged. Multiplying every outcome by k multiplies variance by k² and standard deviation by |k|.

Distinguish an expectation from a realised outcome

An expectation need not be one of the possible outcomes. A model paying either 0 or 10 with equal probabilities has mean 5, although the model never pays exactly 5. The mean summarizes the distribution rather than selecting its next result.

Further references