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Variance and Standard Deviation
Variance is a measure of how far a set of numbers is spread out. The variance ( ) can
be computed as follows:
=
=
Ex) Find the variance of 1, 2, 3, 4, 5.
1
2
3
4
5
Sum = 15
Mean = 3
−2
−1
0
1
2
So the variance will be
4
1
0
1
4
Sum = 10
.
Since the sum of squared deviation,
, can be simplified as
the following formula can also be used for the variance.
Ex) Find the variance of 1, 2, 3, 4 and 5 using the formula above.
Therefore,
=
Standard deviation ( ) is the square root of the variance (
Ex) Find the standard deviation of 1, 2, 3, 4, 5
Since
, standard deviation ( ) =
).
,
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The formula for the variance and the standard deviation above should be used for population. If you have a sample instead of population, you should use the following formula.
sample variance ( ) =
=
sample standard deviation ( ) =
Note that the sum of squared deviation is divided by
variance.
(not
) to find the sample
Ex) Find the variance and standard deviation of the sample: 1, 2, 3, 4, 5
=
=