Calculating statistical mean and standard deviation in one ...
Calculating statistical mean and standard deviation in one pass
Mean formula : sigma(i, 1, n, xi)/n
Standard deviation formula : sqrt(sigma(i, 1, n, (xi-mean)^2)/(n-1))
Let
Sum = sigma(i, 1, n)
Then
Mean = sum / n
Decompose SD
Sqr(sd) = sigma(i, 1, n, (xi-sum/n)^2 / (n-1)
Sqr(sd) = sigma(i, 1, n, ((n*xi-sum)/n)^2 / (n-1)
Sqr(sd) = sigma(i, 1, n, (n*xi-sum)^2 / n^2) / (n-1)
Sqr(sd) = sigma(i, 1, n, (n*xi-sum)^2) / ((n^2) * (n-1))
Sqr(sd) = sigma(i, 1, n, (n*xi)^2-2*(n*x[i])*sum+sum^2)) / ((n^2) * (n-1))
Sqr(sd) = sigma(i, 1, n, (n*xi)^2)-2*n*sum * sigma(i, 1, n, xi) + sigma(i, 1, n, sum^2)) / ((n^2) * (n-1))
Sqr(sd) = ((n^2)*sigma(i, 1, n, (xi)^2) - 2*n*sum^2 + n*sum^2) / ((n^2) * (n-1))
Sqr(sd) = n*(n*sigma(i, 1, n, (xi)^2) - 2 *sum^2 + sum^2) / ((n^2) * (n-1))
Sqr(sd) = (n*sigma(i, 1, n, (xi)^2) - sum^2) / (n*(n-1))
Algorithm
i traversal[1..n]
SUM := SUM + Xi
SUM2 := SUM2 + Xi*Xi
AVERAGE := SUM / n
VARIANCE := (n * SUM2 – SUM * SUM) / (n * ( n – 1))
STDDEV := VARIANCE^(1/2)
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