Algorithms for calculating variance

The formula for calculating variance is:

Variance = [n{x₁² + x₂² + ... + x_n²} - {x₁ + x₂ + ... + x_n}²] / n²-n

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

 
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/