Control Charts for Process Location

16.2 Control Charts for Process Location

  Suppose the quality characteristic of interest is associated with a variable whose observed values result from making measurements. For example, the characteristic might be resistance of electrical wire (ohms), internal diameter of molded rubber expansion joints (cm), or hardness of a certain alloy (Brinell units). One important use of control charts is to see whether some measure of location of the variable’s distribution remains stable over time. The most popular chart for this purpose is the X chart.

  the X chart Based on Known Parameter Values

  Because there is uncertainty about the value of the variable for any particular item or specimen, we denote such a random variable (rv) by X. Assume that for an in- control process, X has a normal distribution with mean value m and standard devia- tion s. Then if X denotes the sample mean for a random sample of size n selected at a particular time point, we know that

  1. E sXd 5 m

  2. s X 5s yÏn

  3. X has a normal distribution. It follows that

  P sm 2 3s X Xm1 3s X d 5 Ps23.00 Z 3.00d 5 .9974

  where Z is a standard normal rv. It is thus highly likely that for an in-control pro-

  cess, the sample mean will fall within 3 standard deviations s3s X d of the process

  mean m.

  Consider first the case in which the values of both m and s are known. Suppose that at each of the time points 1, 2, 3, … , a random sample of size n is

  available. Let x 1 ,x 2 ,x 3 ,…denote the calculated values of the corresponding sam-

  ple means. An X chart results from plotting these x i ’s over time—that is, plotting

  points s1, x 1 d, s2, x 2 d, s3, x 3 d, and so on—and then drawing horizontal lines across

  the plot at

  s

  LCL 5 lower control limit 5 m 2 3 ?

  Ïn s

  UCL 5 upper control limit 5 m 1 3 ?

  Ïn

  The use of charts based on 3 SD limits is traditional, but tradition is certainly not inviolable.

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  Such a plot is often called a 3-sigma chart. Any point outside the control limits suggests that the process may have been out of control at that time, so a search for assignable causes should be initiated.

  ExamplE 16.1 Once each day, three specimens of motor oil are randomly selected from the produc-

  tion process, and each is analyzed to determine viscosity. The accompanying data (Table 16.1) is for a 25-day period. Extensive experience with this process suggests that when the process is in control, viscosity of a specimen is normally distributed

  with mean 10.5 and standard deviation .18. Thus s X 5s yÏn 5 .18yÏ3 5 .104, so

  the 3 SD control limits are

  s

  LCL 5 m 2 3 ?

  5 10.5 2 3 s.104d 5 10.188

  5 10.5 1 3 s.104d 5 10.812

  Ïn

  Table 16.1 Viscosity Data for Example 16.1

  Day

  Viscosity Observations

  x

  s Range

  All points on the control chart shown in Figure 16.2 are between the control limits, indicating stable behavior of the process mean over this time period (the standard deviation and range for each sample will be used in the next subsection).

  16.2 Control Charts for process Location 683

  X chart for the viscosity data of Example 16.1 n

  X charts Based on Estimated Parameters

  In practice it frequently happens that values of m and s are unknown, so they must

  be estimated from sample data prior to determining the control limits. This is espe- cially true when a process is first subjected to a quality control analysis. Denote the number of observations in each sample by n, and let k represent the number of samples available. Typical values of n are 3, 4, 5, or 6; it is recommended that k be at least 20. We assume that the k samples were gathered during a period when the pro- cess was believed to be in control. More will be said about this assumption shortly.

  With x 1 ,x 2 ,…, x k denoting the k calculated sample means, the usual estimate

  of m is simply the average of these means:

  i5 o 1

  k

  x i

  m ˆ5x5 k

  There are two different commonly used methods for estimating s: one based on the k sample standard deviations and the other on the k sample ranges (recall that the sample range is the difference between the largest and smallest sample observa- tions). Prior to the wide availability of good calculators and statistical computer

  software, ease of hand calculation was of paramount consideration, so the range method predominated. However, in the case of a normal population distribution, the unbiased estimator of s based on S is known to have smaller variance than that based on the sample range. Statisticians say that the former estimator is more efficient than the latter. The loss in efficiency for the estimator is slight when n is very small but becomes important for n . 4.

  Recall that the sample standard deviation is not an unbiased estimator for s.

  When X 1 ,…, X n is a random sample from a normal distribution, it can be shown (cf. Exercise 6.37) that

  E sSd 5 a n ?s where

  Ï2Gsny2d

  a n 5 Ïn 2 1G[sn 2 1dy2]

  684 Chapter 16 Quality Control Methods

  and G s?d denotes the gamma function (see Section 4.4). A tabulation of a n for selected n follows:

  n

  a n .886 .921 .940 .952 .959 .965 Let

  k o S i

  i5 S5 1 k

  where S 1 ,S 2 ,…, S k are the sample standard deviations for the k samples. Then

  1 k

  1 k

  K 1 i5 1 2 k i5 1 k o

  1 k

  E sSd 5

  E S i 5 o E o sS i d5 a ?s5 a ?s

  1 a

  S

  n 2 a n a n

  E 5 E sSd 5

  ? a

  n ?s5s

  so sˆ 5 S ya n is an unbiased estimator of s.

  control Limits Based on the Sample Standard deviations

  a n Ïn where

  o x i

  k

  i5 1 o i5 1

  ExamplE 16.2 Referring to the viscosity data of Example 16.1, we had n 5 3 and k 5 25. The

  values of x i and s i si 5 1,…, 25d appear in Table 16.1, from which it follows that

  x5 261.896 y25 5 10.476 and s 5 3.834y25 5 .153. With a 3 5 .886, we have

  These limits differ a bit from previous limits based on m 5 10.5 and s 5 .18 because

  now mˆ 5 10.476 and sˆ 5 s ya 3 5 .173. Inspection of Table 16.1 shows that every x i

  is between these new limits, so again no out-of-control situation is evident.

  n To obtain an estimate of s based on the sample range, note that if X 1 ,…, X n

  form a random sample from a normal distribution, then

  16.2 Control Charts for process Location 685

  R 5 range sX 1 ,…, X n d 5 maxsX 1 ,…, X n d 2 minsX 1 ,…, X n d

  5 max sX 1 2m ,…, X n 2m d 2 minsX 1 2m ,…, X n 2m d

  X 1 2m

  5 1 s

  X n 2m

  X 1 2m

  s 2 1 s

  X n 2m

  s 2 6

  5s max , …,

  min ,…,

  5s? {max sZ 1 ,…, Z n d 2 minsZ 1 ,…, Z n d}

  where Z 1 ,…, Z n are independent standard normal rv’s. Thus

  E sRd 5 s ? Esrange of a standard normal sampled

  5s?b n so that R yb n is an unbiased estimator of s.

  Now denote the ranges for the k samples in the quality control data set by r 1 ,r 2 ,…, r k . The argument just given implies that the estimate

  1 k

  r i

  k o i5 1 r

  comes from an unbiased estimator for s. Selected values of b n appear in the accom- panying table [their computation is based on using statistical theory and numerical

  integration to determine E sminsZ 1 ,…, Z n dd and EsmaxsZ 1 ,…, Z n dd].

  control Limits Based on the Sample ranges r

  where r 5 o k i5 1 r i yk and r 1 ,…, r k are the k individual sample ranges.

  ExamplE 16.3

  Table 16.1 yields r 5 .292, so sˆ 5 .292 yb 3 5 .292 y1.693 5 .172 and

  (Example 16.2 continued)

  These limits are identical to those based on s, and again every x i lies between the limits.

  n

  recomputing control Limits

  We have assumed that the sample data used for estimating m and s was obtained from an in-control process. Suppose, though, that one of the points on the resulting control chart falls outside the control limits. Then if an assignable cause for this out-of-control

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  situation can be found and verified, it is recommended that new control limits be cal- culated after deleting the corresponding sample from the data set. Similarly, if more than one point falls outside the original limits, new limits should be determined after eliminating any such point for which an assignable cause can be identified and dealt with. It may even happen that one or more points fall outside the new limits, in which case the deletionrecomputation process must be repeated.

  Performance characteristics of control charts

  Generally speaking, a control chart will be effective if it gives very few out-of- control signals when the process is in control, but shows a point outside the control limits almost as soon as the process goes out of control. One assessment of a chart’s effectiveness is based on the notion of “error probabilities.” Suppose the variable of interest is normally distributed with known s (the same value for an in-control or out-of-control process). In addition, consider a 3-sigma chart based on the target

  value m 0 , with m 5 m 0 when the process is in control. One error probability is

  a5P sa single sample gives a point outside the control limits when m 5 m 0 d

  5 P sX . m 0 1 3s yÏn or X , m 0 2 3s yÏn when m 5 m 0 d

  1 s yÏn s yÏn

  ,2 3 when m 5 m 0

  The standardized variable Z 5 sX 2 m 0 dyssyÏnd has a standard normal distribution

  when m 5 m 0 , so

  a5 P (Z . 3 or Z , 23) 5 F(23.00) 1 1 2 F(3.00) 5 .0026

  If 3.09 rather than 3 had been used to determine the control limits (this is customary in Great Britain), then

  a5 P (Z . 3.09 or Z , 23.09) 5 .0020

  The use of 3-sigma limits makes it highly unlikely that an out-of-control signal will result from an in-control process.

  Now suppose the process goes out of control because m has shifted to m 1 Ds (D might be positive or negative); D is the number of standard deviations by which m has changed. A second error probability is

  a single sample gives a point inside

  1 the control limits when m 5 m 0 1 Ds 2

  b5 P

  5 P sm 0 2 3s yÏn , X , m 0 1 3s yÏn when m 5 m 0 1 Ds d We now standardize by first subtracting m 0 1 Ds from each term inside the paren-

  theses and then dividing by s yÏn:

  b5 P s23 2 ÏnD , standard normal rv , 3 2 ÏnDd

  5F s3 2 ÏnDd 2 Fs23 2 ÏnDd

  This error probability depends on D, which determines the size of the shift, and on the sample size n. In particular, for fixed D, b will decrease as n increases (the larger the sample size, the more likely it is that an out-of-control signal will result), and for fixed n, b decreases as uDu increases (the larger the magnitude of a shift, the more likely it is that an out-of-control signal will result). The accompanying table gives b for selected values of D when n 5 4.

  16.2 Control Charts for process Location 687

  D .25 .50 .75 1.00

  b when n 5 4 .9936 .9772 .9332 .8413 .5000 .1587 .0668 .0013

  It is clear that a small shift is quite likely to go undetected in a single sample.

  If 3 is replaced by 3.09 in the control limits, then a decreases from .0026 to .002, but for any fixed n and s, b will increase. This is just a manifestation of the inverse relationship between the two types of error probabilities in hypothesis test- ing. For example, changing 3 to 2.5 will increase a and decrease b.

  The error probabilities discussed thus far are computed under the assumption that the variable of interest is normally distributed. If the distribution is only slightly nonnormal, the Central Limit Theorem effect implies that X will have approximately

  a normal distribution even when n is small, in which case the stated error probabili- ties will be approximately correct. This is, of course, no longer the case when the variable’s distribution deviates considerably from normality.

  A second performance assessment involves expected or average run length needed to observe an out-of-control signal. When the process is in control, we should expect to observe many samples before seeing one whose x lies outside the control limits. On the other hand, if a process goes out of control, the expected num- ber of samples necessary to detect this should be small.

  Let p denote the probability that a single sample yields an x value outside the control limits; that is,

  p5P (X , m 0 2 3s yÏn or X . m 0 1 3s yÏn) Consider first an in-control process, so that X 1 ,X 2 ,X 3 ,… are all normally distributed

  with mean value m 0 and standard deviation s yÏn. Define an rv Y by Y5 the first i for which X i falls outside the control limits

  If we think of each sample number as a trial and an out-of-control sample as a success, then Y is the number of (independent) trials necessary to observe a suc- cess. This Y has a geometric distribution, and we showed in Example 3.18 that

  E sYd 5 1yp. The acronym ARL (for average run length) is often used in place of

  E (Y). Because p 5 a for an in-control process, we have

  Replacing 3 in the control limits by 3.09 gives ARL 5 1 y.002 5 500.

  Now suppose that, at a particular time point, the process mean shifts to

  m5m 0 1 Ds . If we define Y to be the first i subsequent to the shift for which a sam- ple generates an out-of-control signal, it is again true that ARL 5 E sYd 5 1yp, but now p 5 1 2 b. The accompanying table gives selected ARLs for a 3-sigma chart when n 5 4. These results again show the chart’s effectiveness in detecting large shifts but also its inability to quickly identify small shifts. When sampling is done rather infrequently, a great many items are likely to be produced before a small shift in m is detected. The CUSUM procedures discussed in Section 16.5 were developed to address this deficiency.

  D .25

  ARL when n 5 4 156.25 43.86 14.97 6.30 2.00 1.19 1.07 1.0013

  688 Chapter 16 Quality Control Methods