Home > Algorithms for calculating variance
In statistics, a formula for calculating the variance of a population of size n is:-
A formula for calculating the unbiased estimation of the population variance from n finite samples is:
-
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)/(5 · 4) = 4.5
- standard deviation =
Note: Details of the variance calculation:
338 = [52 + 72 + 82 + 102 + 102]
40 = [5 + 7 + 8 + 10 + 10]
1 Algorithm
Therefore a simple algorithm to calculate variance can be described by the following pseudocode:
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 = 1 to n
sum += data[i];
sum_sqr += ( data[i] * data[i] );
end for
variance = ((n * sum_sqr) - (sum * sum))/(n*(n-1));
2 Algorithm
Another algorithm which avoids large numbers in sum_sqr while summing up
double avg = 0;
double var = 0;
long n = data.length; // number of elements
for i = 1 to n
avg = (avg*i + data[i]) / (i + 1);
var = (var * (i - 1) + (data[i] - avg)*(data[i] - avg)) / i;
end for
return var; // resulting variance
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