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Uses of the Range Chart

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.

For more information, contact Drs. Robert and Marilyn Hart at robthart@aol.com or (541) 412-0425.