Streaming variance without the sum of squares

The textbook variance formula $\operatorname{Var} = \frac{1}{n}\sum x_i^2 - \bar{x}^2$ is a numerical trap: it forms $\sum x_i^2$, which can overflow a fixed-width integer and, in floating point, subtracts two large nearly-equal numbers, cancelling away your precision.

Compute the running mean and (sample) variance of a stream in a single pass, without ever forming $\sum x_i^2$. Handle the case where values are large but their spread is small.