Streaming variance without storing the data

Question

You receive prices one at a time and must report the running mean and variance at any moment, without keeping the full history in memory. The textbook "average of squares minus square of average" formula loses precision when values are large and close together.

feed 2, 4, 6
mean -> 4.0
population variance -> 2.667   # (4 + 0 + 4) / 3

Maintain the running mean and variance incrementally in O(1) time and space per update.

Accepted Answer

Welford's method keeps three running quantities: the count n, the mean, and M2 (the sum of squared deviations from the current mean). Each new value nudges the mean, and M2 accumulates using the deviation before and after the mean update. Variance is M2 / n (population) or M2 / (n-1) (sample). Tracing 2, 4, 6: the mean moves 2 - 3 - 4, M2 grows 0 - 2 - 8, so the population variance is 8/3 = 2.667. Time O(1) per update, space O(1). Edge cases: variance of zero or one point is 0; sample variance needs at least two points; Welford is far more numerically stable than subtracting two large near-eq…

Streaming variance without storing the data

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