Applied Statistics and Probability for Engineers

16-5XAND ROR SCONTROL CHARTS 609

and we would use as the upper and lower control limits on the Rchart

(16-11)

Setting D 3  1  3 d 3 d 2 and D 4  1  3 d 3 d 2 leads to the following definition.

LCLR

3 d 3

d 2

Ra 1 

3 d 3

d 2

b R

UCLR

3 d 3

d 2

Ra 1 

3 d 3

d 2

b R

UCLc 4  3  21 c 42 (16-13)

LCLc 4  3  21 c^24 CLc 4 

The LCLfor an Rchart can be a negative number. In that case, it is customary to set LCL

to zero. Because the points plotted on an Rchart are nonnegative, no points can fall below an

LCLof zero.

When preliminary samples are used to construct limits for control charts, these limits are

customarily treated as trial values. Therefore, the msample means and ranges should be plotted

on the appropriate charts, and any points that exceed the control limits should be investigated. If

assignable causes for these points are discovered, they should be eliminated and new limits for

the control charts determined. In this way, the process may be eventually brought into statistical

control and its inherent capabilities assessed. Other changes in process centering and dispersion

may then be contemplated. Also, we often study the Rchart first because if the process variabil-

ity is not constant over time the control limits calculated for the chart can be misleading.

Rather than base control charts on ranges, a more modern approach is to calculate the

standard deviation of each subgroup and plot these standard deviations to monitor the process

standard deviation . This is called an Schart. When an Schart is used, it is common to use

these standard deviations to develop control limits for the chart. Typically, the sample size

used for subgroups is small (fewer than 10) and in that case there is usually little difference in

the chart generated from ranges or standard deviations. However, because computer soft-

ware is often used to implement control charts, Scharts are quite common. Details to construct

these charts follow.

In Section 7-2.2 on the CD, it was shown that Sis a biased estimator of . That is,

E(S)c 4 where c 4 is a constant that is near, but not equal to, 1. Furthermore, a calculation

similar to the one used for E(S) can derive the standard deviation of the statistic Swith the re-

sult  21 c^24. Therefore, the center line and three-sigma control limits for Sare

X

X

X

The center line and upper and lower control limits for an Rchart are

(16-12)

where is the sample average range, and the constants D 3 and D 4 are tabulated for

various sample sizes in Appendix Table X.

r

UCLD 4 r CLr LCLD 3 r

RChart

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