Robert F. Hart, Ph. D.
Marilyn K. Hart, Ph.D.
Introduction
All quality control professionals are familiar
with the range (R) chart as a control chart
to monitor the variability of the product. Actually,
in its normal use, the R chart monitors the
total variability of the product and
the measurement system. A second use of the
R chart is to monitor the variability if the
measurement system and knowing that, the actual
variability of just the product can be found.
This paper discusses these two uses of the R
chart.
The Usual R Chart
To make the usual R chart, subgroups of size
n (let's say of size n = 4) are taken. That
means each time a subgroup is taken, four different
pieces of product are measured.
These four pieces of product are usually four
sequential pieces so that the variability between
them can be kept to a minimum, yielding the
tightest control limits. This will allow the
control limits to most effectively detect special-cause
variation if it is present.
One way to then estimate the total variability
in the product and measurement process is by
the formula
(Eq.1)
where 
is
the estimate of the population standard deviation
for the total variability and d2
is the factor that depends on the subgroup size
and is found from a table, an abbreviated one
given in Table 1.
Table 1. Table of d2 Factors
Subgroup size n
|
d2
|
|
2
|
1.13
|
|
3
|
1.69
|
|
4
|
2.06
|
|
5
|
2.33
|
|
6
|
2.53
|
|
7
|
2.70
|
|
8
|
2.85
|
|
9
|
2.97
|
|
10
|
3.08
|
Say twenty five subgroups of size n = 4 are
taken for the usual control chart. The Xbar
and the R chart are both in control. The R chart
is displayed in Figure 1. Note that 
=
5.6 so the total variability (
) for the combined product and measurement system
is, by a variation of Equation 1
.
Figure 1. R Chart on 25 Subgroups of Product
But everything varies, including measurements.
To find out how much the measurement system
varies, the same operator using the same gauge
under the same conditions measures 40 parts,
each twice. These measurements must be independent,
so with the help of an assistant the parts are
numbered and are given to the operator in a
random order so that the first measurement is
not remembered. The two measurements for the
first piece are recorded together and the range
computed. Similarly, the two measurements for
the second piece are recorded together and the
range computed, and so on for all 40 pieces.
The R chart is in control (Figure 2).
Figure 2. R Chart on Repeat Measurements From
40 Pieces of Product
Here =
1.7. With this sampling scheme, the standard
deviation estimate is by using a variation of
Equation 1.
The
emphasizes that this is the standard deviation
of the measurement system. To find out how much
variability is really only due to product, it
is noted that
(Eq. 2)
To find 
we substitute into Equation 2 and solve for
.
= 2.2.
Conclusion
It is important to remember the two uses of
an R control chart. One is to monitor the variability
of product (combined with the measurement variation)
and the other is to study the measuring variability
by itself, thereby being able to discover how
much of the original variability was due to
product.
