Page 440 - Design for Six Sigma for Service (Six SIGMA Operational Methods)

398 Chapter Eleven

The range is very easy to compute, but it only gives the distance of the two
most extreme observations. It is not a good measure of variation for the
whole data set. The variance and standard deviation are better measures in
this aspect.

Variance
The variance of a sample of n measurements y , y ,…,y is defined as
2
1
n
n
s = 1 ∑ ( y − y) 2 (11.4)
2
n 1− i 1 i
=
Example 11.5
For the data of Example 11.1, the variance can be computed as
−
−
s = 1 [( 547 572 02 + ( 563 572 02) 2
2
2
)
.
.
−
47 1
−
2
.
+⋅⋅⋅+ ( 575 572 02) ] = 601.72
2
Sample variance s is obviously an average of the sum of squared deviations
from the mean of all observations. Squared deviation makes sense because
no matter if an observation is smaller or larger than the mean, the squared
deviation will always be positive. The average of the squared deviation is a
measure of variation for the whole data set. However, the numerical scale
and measurement unit of variance is the square of the original data. For
example, if the original data is length in inches, the variance will be in the
unit of squared inch, which cannot compare well with the original data.
Standard Deviation
The standard deviation is the square root of variance, specifically, the standard
deviation of a sample of n measurements y , y ,…,y and is defined as
1
n
2
n
s = s = 1 ∑ ( y − y) 2 (11.5)
2
i
n −1
` i=1
Example 11.6
For the data of Example 11.1, the standard deviation can be computed as
s = s = 601 72 = 24 53
2
.
.
Measure of Relative Standing
The measure of relative standing provides a numerical value or score that
describes a predefined location relative to other observations in a data set.