E1C04 09/14/2010
14:7:42 Page 134
SOLUTION The sample mean value is computed for the N ¼ 20 values by the relation
x ¼
1
20
X 20
i¼1
x i ¼ 1:02
This, in turn, is used to compute the sample standard deviation
s x ¼
1
19
X 20
i¼1
x i À x
ð
Þ
2
"
# 1=2
¼ 0:16
The degrees of freedom in the data set is n ¼ N À 1 ¼ 19. From Table 4.4 at 95% probability, t 19,95 ¼
2.093. Then, the interval of values in which 95% of the measurements of x should lie is given by
Equation 4.15:
x i ¼ 1:02 Æ ð2:093 Â 0:16Þ ¼ 1:02 Æ 0:33 ð95%Þ
Accordingly, if a 21st data point were taken, there is a 95% probability that its value would lie
between 0.69 and 1.35.
The true mean value is estimated by the sample mean value. However, the random uncertainty at
95% probability for this estimate is t 19;95 s x , where
s x ¼
s x
ffiffiffiffi
N
p ¼
0:16
ffiffiffiffiffi
20
p ¼ 0:036 % 0:04
Then, in the absence of systematic errors, we write, from Equation 4.19,
x
0
¼ x Æ t 19;95 s x ¼ 1:02 Æ 0:08 ð95%Þ
So at 95% confidence, the true mean value lies between 0.94 and 1.10. Program Finite-population.vi
demonstrates the effect of sample size on the histogram and the statistics of the data set.
Pooled Statistics
As discussed in Chapter 1, a good test plan uses duplicate tests or replication, as well as repetition.
Since replications are independent estimates of the same measured value, their data represent
separate data samples that can be combined to provide a better statistical estimate of a measured
variable than are obtained from a single sampling. Samplings that are grouped in a manner so as to
determine a common set of statistics are said to be pooled.
Consider M replicates of a measurement of variable x, each of N repeated readings so as to yield
the data set x ij , where i ¼ 1, 2, . . . , N and j ¼ 1, 2, . . . , M. The pooled mean of x is defined by
hxi ¼
1
MN
X M
j¼1
X N
i¼1
x ij
ð4:20Þ
The pooled standard deviation of x is defined by
hs x i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
M N À 1
ð
Þ
X M
j¼1
X N
i¼1
x ij À x j
2
v
u
u
t
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
M
X M
j¼1
s 2
x j
v
u
u
t
ð4:21Þ
134 Chapter 4 Probability and Statistics
14:7:42 Page 134
SOLUTION The sample mean value is computed for the N ¼ 20 values by the relation
x ¼
1
20
X 20
i¼1
x i ¼ 1:02
This, in turn, is used to compute the sample standard deviation
s x ¼
1
19
X 20
i¼1
x i À x
ð
Þ
2
"
# 1=2
¼ 0:16
The degrees of freedom in the data set is n ¼ N À 1 ¼ 19. From Table 4.4 at 95% probability, t 19,95 ¼
2.093. Then, the interval of values in which 95% of the measurements of x should lie is given by
Equation 4.15:
x i ¼ 1:02 Æ ð2:093 Â 0:16Þ ¼ 1:02 Æ 0:33 ð95%Þ
Accordingly, if a 21st data point were taken, there is a 95% probability that its value would lie
between 0.69 and 1.35.
The true mean value is estimated by the sample mean value. However, the random uncertainty at
95% probability for this estimate is t 19;95 s x , where
s x ¼
s x
ffiffiffiffi
N
p ¼
0:16
ffiffiffiffiffi
20
p ¼ 0:036 % 0:04
Then, in the absence of systematic errors, we write, from Equation 4.19,
x
0
¼ x Æ t 19;95 s x ¼ 1:02 Æ 0:08 ð95%Þ
So at 95% confidence, the true mean value lies between 0.94 and 1.10. Program Finite-population.vi
demonstrates the effect of sample size on the histogram and the statistics of the data set.
Pooled Statistics
As discussed in Chapter 1, a good test plan uses duplicate tests or replication, as well as repetition.
Since replications are independent estimates of the same measured value, their data represent
separate data samples that can be combined to provide a better statistical estimate of a measured
variable than are obtained from a single sampling. Samplings that are grouped in a manner so as to
determine a common set of statistics are said to be pooled.
Consider M replicates of a measurement of variable x, each of N repeated readings so as to yield
the data set x ij , where i ¼ 1, 2, . . . , N and j ¼ 1, 2, . . . , M. The pooled mean of x is defined by
hxi ¼
1
MN
X M
j¼1
X N
i¼1
x ij
ð4:20Þ
The pooled standard deviation of x is defined by
hs x i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
M N À 1
ð
Þ
X M
j¼1
X N
i¼1
x ij À x j
2
v
u
u
t
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
M
X M
j¼1
s 2
x j
v
u
u
t
ð4:21Þ
134 Chapter 4 Probability and Statistics