### Mean and Standard Deviation

 Algorithms  glossary  Numerical   Num'Errors   Polynomials   Stirling   Mean&S.D.   Integration

The mean, &mu, of N numbers, A[1], ..., A[N], is their sum divided by N, i.e. `(∑1..N A[i])/N`. Their variance is `(∑1..N (A[i]-μ)2)/N` and their standard deviation, σ, is `sqrt(variance)`; note that these quantities are always >=0. The mean gives the "centre of gravity" (CG) of the numbers, and the standard deviation indicates how far they stray from the CG, on average.

Both the mean and the standard deviation can be calculated on a single scan through A[ ] even though the mean is not known until the end of the scan:

variance
= σ2
= ( ∑i=1..N (A[i]-μ)2 ) / N
= ( (∑ A[i]2) - 2*μ*(∑ A[i]) + N*μ2 ) / N
= ( (∑ A[i]2) - 2*μ*sum ) / N + μ2
= ( ∑i=1..N A[i]2 ) / N - μ2
i.e. the mean squares minus the squared mean.
Hence `σ = sqrt(sumSq / N - μ2)`
where `sumSq = ∑1..N A[i]2`

This gives the following algorithm:

```  sum := 0.0;
sumSq := 0.0;

for i in {1 .. N} do
sum +:= A[i];
sumSq +:= A[i]2
end for;

mean := sum / N;
stdDev := sqrt(sumSq / N - mean2);
```

inp[]
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-- L.A., 1999

#### Notes

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 © L. Allison   http://www.allisons.org/ll/   (or as otherwise indicated), Faculty of Information Technology (Clayton), Monash University, Australia 3800 (6/'05 was School of Computer Science and Software Engineering, Fac. Info. Tech., Monash University, was Department of Computer Science, Fac. Comp. & Info. Tech., '89 was Department of Computer Science, Fac. Sci., '68-'71 was Department of Information Science, Fac. Sci.) Created with "vi (Linux + Solaris)",  charset=iso-8859-1,  fetched Thursday, 27-Oct-2016 19:56:29 EST.