Expected value, variance and standard deviation

In short

• The expected value $E(X)$ multiplies each possible outcome by its probability and adds the products.
• It is a forward-looking forecast (or the true population mean), unlike a sample mean, which equally weights past observations.
• Variance is the probability-weighted average of squared deviations from $E(X)$. It is never negative and is zero only when the outcome is certain.
• Standard deviation is the positive square root of variance and is in the same units as the variable, so it is easier to interpret.
• Always work in the same order: expected value, then variance, then standard deviation.