Control Charts for Attributes

16.4 Control Charts for Attributes

  The term attribute data is used in the quality control literature to describe two

  situations:

  1. Each item produced is either defective or nondefective (conforms to specifica- tions or does not).

  2. A single item may have one or more defects, and the number of defects is determined.

  In the former case, a control chart is based on the binomial distribution; in the latter case, the Poisson distribution is the basis for a chart.

  696 Chapter 16 Quality Control Methods

  the p chart for Fraction defective

  Suppose that when a process is in control, the probability that any particular item is defective is p (equivalently, p is the long-run proportion of defective items for an in-control process) and that different items are independent of one another with respect to their conditions. Consider a sample of n items obtained at a particular time, and let X be the number of defectives and pˆ 5 X yn. Because X has a binomial distribution, E sXd 5 np and VsXd 5 nps1 2 pd, so

  spˆd 5 n

  p s1 2 pd

  V

  E spˆd 5 p

  Also, if np 10 and n s1 2 pd 10, pˆ has approximately a normal distribution.

  In the case of known p (or a chart based on target value), the control limits are

  Î n Î n

  If each sample consists of n items, the number of defective items in the ith sample is

  x i , and pˆ i 5 x i yn, then p ˆ , pˆ 1 2 , pˆ 3 ,… are plotted on the control chart.

  Usually the value of p must be estimated from the data. Suppose that k samples from what is believed to be an in-control process are available, and let

  k o pˆ i

  i5 p5 1 k

  The estimate p is then used in place of p in the aforementioned control limits.

  The p chart for the fraction of defective items has its center line at height p and control limits

  Î n

  p (1 2 p) LCL 5 p 2 3

  Î n

  p (1 2 p) UCL 5 p 1 3

  If LCL is negative, it is replaced by 0.

  ExamplE 16.6

  A sample of 100 cups from a particular dinnerware pattern was selected on each of

  25 successive days, and each was examined for defects. The resulting numbers of unacceptable cups and corresponding sample proportions are as follows:

  Day (i)

  1 2 3 4 5 6 7 8 9 10 11 12 13 x i 743649675378 4 pˆ i .07 .04 .03 .06 .04 .09 .06 .07 .05 .03 .07 .08 .04

  Day (i) 14 15 16 17 18 19 20 21 22 23 24 25

  x i 6 2 9 7 6 7 11 6 7 4 8 6 pˆ i .06 .02 .09 .07 .06 .07 .11 .06 .07 .04 .08 .06

  16.4 Control Charts for attributes 697

  Assuming that the process was in control during this period, let’s establish control

  limits and construct a p chart. Sin ce opˆ i 5 1.52, p 5 1.52 y25 5 .0608 and

  LCL 5 .0608 2 3 Ï s.0608ds.9392dy100 5 .0608 2 .0717 5 2.0109 UCL 5 .0608 1 3 Ï s.0608ds.9392dy100 5 .0608 1 .0717 5 .1325

  The LCL is therefore set at 0. The chart pictured in Figure 16.5 shows that all points are within the control limits. This is consistent with an in-control process.

  Figure 16.5 Control chart for fraction-defective data of Example 16.6 n

  the c chart for number of defectives

  We now consider situations in which the observation at each time point is the number of defects in a unit of some sort. The unit may consist of a single item (e.g., one auto- mobile) or a group of items (e.g., blemishes on a set of four tires). In the second case, the group size is assumed to be the same at each time point.

  The control chart for number of defectives is based on the Poisson probability distribution. Recall that if Y is a Poisson random variable with parameter m, then

  sYd 5 m

  Y

  Also, Y has approximately a normal distribution when m is large (m 10 will suffice

  for most purposes). Furthermore, if Y 1 ,Y 2 ,…, Y n are independent Poisson variables with parameters m 1 ,m 2 ,…, m n , it can be shown that Y 1 1…1 Y n has a Poisson distribution with parameter m 1 1…1m n . In particular, if m 1 5…5m n 5m (the

  distribution of the number of defects per item is the same for each item), then the Poisson parameter is nm.

  Let m denote the Poisson parameter for the number of defects in a unit (it is the expected number of defects per unit). In the case of known m (or a chart based on a target value),

  Ïm

  LCL 5 m 2 3 Ïm

  UCL 5 m 1 3

  With x i denoting the total number of defects in the ith unit si 5 1, 2, 3,…d, then

  points at heights x 1 ,x 2 ,x 3 ,… are plotted on the chart. Usually the value of m must

  be estimated from the data. Since E sX i d 5 m, it is natural to use the estimate m 5 x

  (based on x 1 ,x 2 ,…, x k ).

  698 Chapter 16 Quality Control Methods

  The c chart for the number of defectives in a unit has center line at x and

  LCL 5 x 2 3 Ïx UCL 5 x 1 3 Ïx

  If LCL is negative, it is replaced by 0.

  ExamplE 16.7 A company manufactures metal panels that are baked after first being coated with a slurry of powdered ceramic. Flaws sometimes appear in the finish of these panels, and the company wishes to establish a control chart for the number of flaws. The number of flaws in each of the 24 panels sampled at regular time intervals are as follows:

  Ï9.79 5 19.18

  with ox i 5 235 and mˆ 5 x 5 235 y24 5 9.79. The control limits are LCL 5 9.79 2 3 Ï9.79 5 .40

  UCL 5 9.79 1 3

  The control chart is in Figure 16.6. The point corresponding to the fifteenth panel lies above the UCL. Upon investigation, the slurry used on that panel was discovered to be of unusually low viscosity (an assignable cause). Eliminating that observation gives x 5 214 y23 5 9.30 and new control limits

  Ï9.30 5 18.45

  LCL 5 9.30 2 3 Ï9.30 5 .15

  UCL 5 9.30 1 3

  x

  Original UCL Final UCL

  5 Original LCL Final LCL

  Sample 0 5 10 15 20 number

  Figure 16.6 Control chart for number of flaws data of Example 16.7

  The remaining 23 observations all lie between these limits, indicating an in-control process.

  n

  control charts Based on transformed data

  The use of 3-sigma control limits is presumed to result in P sstatistic , LCLd < P sstatistic . UCLd < .0013 when the process is in control. However, when p is small, the normal approximation to the distribution of pˆ 5 X yn will often not be very accurate in the extreme tails. Table 16.3 gives evidence of this behavior for

  16.4 Control Charts for attributes 699

  Table 16.3 In-Control Probabilities for a p Chart

  p

  n

  P(pˆ , LCL)

  P( pˆ . UCL)

  P(out-of-control point)

  selected values of p and n (the value of p is used to calculate the control limits). In many cases, the probability that a single point falls outside the control limits is very different from the nominal probability of .0026.

  This problem can be remedied by applying a transformation to the data. Let

  h (X) denote a function applied to transform the binomial variable X. Then h(?) should be chosen so that h(X) has approximately a normal distribution and this approximation is accurate in the tails. A recommended transformation is based on

  the arcsin (i.e., sin 2 1 ) function: Y5h sXd 5 sin 2 1 sÏXynd

  Then Y is approximately normal with mean value sin 2 1 sÏpd and variance 1y(4n); note that the variance is independent of p. Let y i 5 sin 2 1 sÏx i ynd. Then points on the control chart are at heights y 1 ,y 2 ,…. For known n, the control limits are

  sÏpd 1 3Ï1ys4nd

  LCL 5 sin 2 1 sÏpd 2 3Ï1ys4nd

  UCL 5 sin 2 1

  When p is not known, sin 2 1 ( Ïp) is replaced by y.

  Similar comments apply to the Poisson distribution when m is small. The suggested transformation is Y 5 h sXd 5 2ÏX, which has mean value 2Ïm and vari- ance 1. Resulting control limits are 2 Ïm 6 3 when m is known and y 6 3 otherwise. The book Statistical Methods for Quality Improvement listed in the chapter bibliography discusses these issues in greater detail.

  EXERCISES Section 16.4 (21–28)

  21. On each of the previous 25 days, 100 electronic devices

  of a certain type were randomly selected and subjected to

  20, 17, 18, 12, 24, 30, 16, 11, 20, 14, 28. Construct a

  a severe heat stress test. The total number of items that

  p chart and examine it for any out-of-control points.

  failed to pass the test was 578.

  23. When n 5 150, what is the smallest value of p for which

  a. Determine control limits for a 3-sigma p chart.

  the LCL in a p chart is positive?

  b. The highest number of failed items on a given day was 39, and the lowest number was 13. Does either of

  24. Refer to the data of Exercise 22, and construct a control

  these correspond to an out-of-control point? Explain.

  chart using the sin 2 1 transformation as suggested in

  the text.

  22. A sample of 200 ROM computer chips was selected on each of 30 consecutive days, and the number of noncon-

  25. The accompanying observations are numbers of defects in

  forming chips on each day was as follows: 10, 18, 24, 17,

  25 1-square-yard specimens of woven fabric of a certain

  700 Chapter 16 Quality Control Methods

  1, 5, 4, 6. Construct a c chart for the number of defects.

  Panel Examined Blemishes

  26. For what x values will the LCL in a c chart be negative?

  27. In some situations, the sizes of sampled specimens vary,

  and larger specimens are expected to have more defects

  than smaller ones. For example, sizes of fabric samples

  inspected for flaws might vary over time. Alternatively,

  5 1.0 5 the number of items inspected might change with 6 1.0 5

  time. Let

  the number of defects observed at time i

  size of entity inspected at time i

  where “size” might refer to area, length, volume, or

  15 1.0 6 simply the number of items inspected. Then a u chart 16 1.0 12

  plots u 1 ,u 2 ,…, has center line u, and the control limits

  for the ith observations are u 6 3 Ïuyg i . 18 .6

  Painted panels were examined in time sequence,

  and for each one, the number of blemishes in a speci-

  fied sampling region was determined. The surface area

  (ft 2 ) of the region examined varied from panel to panel.

  28. Construct a control chart for the data of Exercise 25 by

  Results are given below. Construct a u chart.

  using the transformation suggested in the text.