Algorithms for calculating variance

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The formula for calculating the population variance:

The formula for calculating unbiased estimation of the population variance from finite samples is:

The method of calculation may be more easily understood from the table below where the mean is 8.

i	x_i	x_i-mean	(x_i-mean)²
(index)	(datum)	(deviation)	(squared deviation)
1	5	-3	9
2	7	-1	1
3	8	0	0
4	10	2	4
5	10	2	4
n=5	sum=40	0	18

 mean = 40/5 = 8

 variance = (5*338 - 40²)/(25-5) = 4.5

 standard deviation = = 2.12

Note: Details of the variance calculation:

338 = [5² + 7² + 8² + 10² + 10²]
40 = [5 + 7 + 8 + 10 + 10]

Algorithm

Therefore a simple algorithm to calculate variance can be:

double sum;
double sum_sqr;
double variance;
long n = data.length; // the number of elements in the data array (the actual syntax is language-specific)
for i = 0 to n
sum += data[i];

sum_sqr += ( data[i] * data[i] );

end for
variance = ((n * sum_sqr) - (sum * sum))/(n*(n-1));

Algorithm

Another algorithm which avoids large numbers in sum_sqr while summing up

double avg;
double var;
long n = data.length; // number of elements
for i = 0 to n
avg = (avg*i + data[i]) / (i + 1);

if (i > 0) var += (var * (i - 1) + (x - avg)*(x - avg)) / i;

end for
return var; // resulting variance