Page 232 - Applied Statistics And Probability For Engineers

c06.qxd 5/14/02 9:54 M Page 193 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:MONTGOMERY:REVISES UPLO D CH114 FIN L:Quark Files:

6-1 DATA SUMMARY AND DISPLAY 193

numerical accuracy. A more efficient computational formula for the sample variance is
obtained as follows:
n n n n
a 1x x2 2 a 1x i x 2xx 2 a x i nx 2x a x i
2
2
2
2
i
i
2
s i 1 i 1 i 1 i 1
n 1 n 1 n 1
n
and since x 11 n2 g i 1 x , this last equation reduces to
i
n 2
a a x b
n i
2
a x i i 1 n
2
s i 1 (6-4)
n 1
Note that Equation 6-4 requires squaring each individual x , then squaring the sum of the x , i
i
2
2
subtracting 1 g x 2 n from g x i , and finally dividing by n 1. Sometimes this is called the
i
shortcut method for calculating s 2 (or s).
EXAMPLE 6-3 We will calculate the sample variance and standard deviation using the shortcut method,
Equation 6-4. The formula gives
n 2
a a x b
n i 1 i 11042 2
2
a x i n 1353.6
2
s i 1 8 1.60 0.2286 1pounds2 2
n 1 7 7
and
s 10.2286 0.48 pounds

These results agree exactly with those obtained previously.

2
s
Analogous to the sample variance , the variability in the population is defined by the
2

population variance ( ). As in earlier chapters, the positive square root of 2 , or , will
denote the population standard deviation. When the population is finite and consists of N
values, we may define the population variance as
N
a 1x 2 2
i
2
i 1 (6-5)
N
We observed previously that the sample mean could be used as an estimate of the population
mean. Similarly, the sample variance is an estimate of the population variance. In Chapter 7,
we will discuss estimation of parameters more formally.
Note that the divisor for the sample variance is the sample size minus one 1n 12, while
for the population variance it is the population size N. If we knew the true value of the popu-
lation mean , we could find the sample variance as the average squared deviation of the sam-
ple observations about . In practice, the value of is almost never known, and so the sum of