Wikipedia: Algorithms for calculating variance

Difference (from prior major revision) (minor diff, author diff)

Changed: 3c3

*Variance = {(x₁-µ)² + (x₂-µ)² + ... + (x_n-µ)²} / n; where µ is the Arithmetic Mean of the data set.

*Variance = {(x₁-µ)² + (x₂-µ)² + ... + (x_n-µ)²} / n; where µ is the arithmetic mean of the data set.

Removed: 11,19d10

:Editorial comment: it is actually the first formula that has precision problems when dealing with limited-precision arithmetic. If the difference between measurements and the mean is very small, then the first formula will yield precision problems, as information will be lost in the (x_i - µ) operation. There is no such loss of significance in the intermediate operations of the second formula. -- ScottMoonen

:Editorial comment the second: in fact, the second formula is the one more commonly beset with problems. The first can have problems when the mean is very large relative to the variance, but this is relatively rare in practice and this problem also affects the second formula. Much more common is the situation where you have comparable mean and variance and a very large number of observations. In this case, the second formula will result in the subtraction of two very large numbers whose difference is relatively small (by a factor roughly equal to the number of observations). If you have one million observations, you lose roughly six significant figures with the second formula if you use ordinary floating point arithmetic. -- TedDunning

:''comment: The problem may occur
#when the deviations are very small relative to the mean or
#when they are small relative to the representational capacity of the arithmetic instrument (floating point computer, fixed point calculator, paper and pencil).
To be precise we have to specify the instrument and the nature of the data. -- DickBeldin''

Changed: 42c33,35

back to Variance

back to Variance

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The variance of a population is defined as the root mean squared deviation from the mean. That mouthful says the same as the formula below.

Variance = {(x₁-µ)² + (x₂-µ)² + ... + (x_n-µ)²} / n; where µ is the arithmetic mean of the data set.

There is another formula for calculating variance which you may see. It uses the sum of all the data and the sum of the squares. The formula is:

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

This formula was introduced when the prevailing calculators made it much easier to sum squares and the raw data than to sum the squared deviations. Because this formula can result in loss of precision, it should no longer be recommended except for small exercises.

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 = 18/5 = 3.6
standard deviation = 1.897366596101 or 1.9

Note that the column of deviations sums to zero. This is always the case. Note also that we round the standard deviation to one more than the number of significant digits in the mean.

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