E1C04 09/14/2010
14:7:42 Page 135
with degrees of freedom, n ¼ M N À 1
ð
Þ. The pooled standard deviation of the means of x is
defined by
hs x i ¼
hs x i
ffiffiffiffiffiffiffiffi
MN
p
ð4:22Þ
When the number of measurements of x are not the same between replications, then it is appropriate
to weight each replication by its particular degrees of freedom. The pooled mean is then defined by
its weighted mean
hxi ¼
P M
j¼1
N j x j
P M
j¼1
N j
ð4:23Þ
where subscript j refers to a particular data set. The pooled standard deviation is given by
hs x i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
n 1 s 2
x 1
þ n 2 s 2
x 2
þ Á Á Á þ n M s 2
x M
n 1 þ n 2 þ Á Á Á þ n M
s
ð4:24Þ
with degrees of freedom n ¼
P M
j¼1 n j ¼
P M
j¼1 N j À 1
À
Á
, and the pooled standard deviation of the
means is given by
s x
À Á ¼
hs x i
ffiffiffiffiffiffiffiffiffiffiffi
P M
j¼1
N j
s
ð4:25Þ
4.5 CHI-SQUARED DISTRIBUTION
We have already discussed how different finite-sized data sets of the same measured variable would
have somewhat different statistics. We used this argument to develop the concept of the standard
deviation of the means as a precision indicator in the mean value. Similarly, we can estimate how
well s
2
x predicts s
2 . If we plotted the sample standard deviation for many data sets, each having N
data points, we would generate the probability density function, p(x
2 ). The p(x
2 ) follows the
so-called chi-squared (x
2 ) distribution depicted in Figure 4.7.
p(χ 2 )
v = 1
v = 3
v = 10
χ 2
Figure 4.7 The x
2 distribution with
its dependency on degrees of
freedom.
4.5 Chi-Squared Distribution 135