Expected Value

Written $E[X]$, the expected value is $\sum_i x_i P(x_i)$: multiply each possible value by its probability and add. For a plain dataset every point is equally likely, so this reduces to the ordinary arithmetic mean.

It need not be a value the variable can actually take. A coin with a 0.3 chance of heads, scored 1 for heads and 0 for tails, has an expected value of 0.3 — an outcome that never occurs. "Expected" means the long-run average, not the likely result.

The property that carries the most weight is linearity: $E[X + Y] = E[X] + E[Y]$ holds whether or not $X$ and $Y$ are related, and $E[aX + b] = aE[X] + b$. That unconditional additivity is why expectations are tractable while variances are not — $\text{Var}(X + Y)$ needs a covariance term, and expectation needs nothing.