Re: variance
- From: rob@xxxxxxxxxxxxxx (Rob Johnson)
- Date: Thu, 16 Mar 2006 02:24:45 GMT
In article <1142472447.227577.326260@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
gmarkowsky@xxxxxxxxx wrote:
Can someone please tell me why sometimes when calculating the variance
of a sample wee divide by n-1 instead of n? Thanks.
The variance of a discrete distribution is defined as
n n
1 --- _ 2 1 --- 2 _ 2
- > (x - x) = - > x - x
n --- k n --- k
k=1 k=1
That is, the mean of the squares minus the square of the mean.
However, when taking sampling, in general the sample mean differs
from the distribution mean. The sample mean is shifted toward
the samples which, in general, causes the sample variance to be
smaller than the distribution variance. As shown on the page at
<http://www.whim.org/nebula/math/varn-1.html>, the the expected
sample variance is (n-1)/n times the distribution variance.
Therefore, the proper way to correct this is to multiply the
sample variance by n/(n-1). Since the formula for the variance
has n in the denominator, the correction replaces the n with an
n-1.
Rob Johnson <rob@xxxxxxxxxxxxxx>
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