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# Algorithms for calculating variance

The formula for calculating variance is:

Variance = [n{x12 + x22 + ... + xn2} - {x1 + x2 + ... + xn}2] / n2-n

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

 i xi xi-mean (xi-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 = SQUARE ROOT [ VARIANCE ] = 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));

Corrected by: Salman Quazi [salman at ucla dot edu]
http://www.setcomputing.com/

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