Sample Variance Computation
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المرجع الالكتروني للمعلوماتيه
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24-2-2021
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Sample Variance Computation
When computing the sample variance 
numerically, the mean must be computed before 
can be determined. This requires storing the set of sample values. However, it is possible to calculate 
using a recursion relationship involving only the last sample as follows. This means 
itself need not be precomputed, and only a running set of values need be stored at each step.
In the following, use the somewhat less than optimal notation 
to denote 
calculated from the first 
samples (i.e., not the 
th moment)
 |
(1)
|
and let 
denotes the value for the bias-corrected sample variance 
calculated from the first 
samples. The first few values calculated for the mean are
Therefore, for 
, 3 it is true that
 |
(5)
|
Therefore, by induction,
By the definition of the sample variance,
 |
(10)
|
for 
. Defining 
, 
can then be computed using the recurrence equation
Working on the first term,
Use (◇) to write
 |
(17)
|
so
 |
(18)
|
Now work on the second term in (◇),
 |
(19)
|
Considering the third term in (◇),
But
 |
(23)
|
so
Finally, plugging (◇), (◇), and (◇) into (◇),
gives the desired expression for 
in terms of 
, 
, and 
,
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