CONTROL CHARTS FOR ATTRIBUTE VARIABLES
Like the continuous variable control charts, the control chart for an attribute variable also takes the form of a sideways, two-way comparison of a two-sided hypothesis test. The difference between the continuous and attribute control charts lie in the underlying distributions. Most continuous variables are well represented by the Normal Distribution, while the attribute variables are typically modeled by the Binomial or Poisson Distributions.
Both the Binomial and Poisson are forms of discrete distributions, which means that the variable takes on non-negative, integer values (such as defect counts). Continuous distributions, on the other hand, allow the variable to take on non-negative, real values, such as length measurements. For both types of distributions, the probability of observing a particular outcome was found using the probability density function (the curve) of the distribution.
With continuous distributions, the probability of observing a range of values was defined by the area under the curve. This area was computed by integration (or looking up the value from a table of integral values). For a discrete distribution, the probability of observing a range of outcomes is found by summing up the probability of observing each outcome in the range of values.
Example:
Assume that you have two six-sided dice. The possible outcomes for the sum of the two die are the discrete values from 2 through 12. If you tabulated the different ways (rolls of each die) in which you could reach each of the totals, you would have a histogram that describes the PDF for your dice (assuming that they are “fair”). Create the histogram and use this knowledge to answer the questions below:
What is the most frequently occurring sum that you could roll?
What is the probability of obtaining the most likely sum in a single roll of the dice?
What is the probability of obtaining a sum greater than 2 and less than 11?
08105240d77bcb3a747aa9a4451645f3.doc Page 1 of 8 There are four types of control charts commonly used with attribute data. The decision on which to use depends on: (a) whether or not a unit is to be classified defective (having one or more defects), or if the number of defects in a unit (or per unit) is of interest; and (b) if the size of the rational sampling group is fixed or variable.
A unit can be classified as defective (or non-conforming) if it contains one or more defects. If the number of defective units in a sample of units is of interest, and if the number of units in the sample is constant, the np-Chart is used to track the production process. However, if the number of units in the sample varies, and the interest in the fraction (or percentage) of defective units in the sample, then the p-Chart is used.
Sometimes just the number of defects per inspection unit is a better measure of performance. If it is easier to count the number of defects in a fixed-size inspection unit (defects per 100 solder joints), then the c-Chart is used. But if the size of the inspection unit could vary (perhaps the current inspection unit has 350 solder joints this time), then the u-Chart will let us track the number of defects on a per-unit basis (where the number of inspection units is 3.5 in this case).
Figure 1 (below) depicts the decision process for choosing the most appropriate control chart. The following sections describe how the control limits for these control charts are computed, and how these charts are interpreted.
P-Charts
P-Charts are derived from the Binomial Distribution, and are used to track the proportion (p) that are defective within a variable sample size. If D is the number of defective units in a random sample of size n, then our sample proportion defective will be:
D pˆ n
Since these samples come from a binomial distribution, and assuming that we knew the true proportion defective in all the product was p, then the probability that the number of defectives (D) in a sample of size n is exactly x units is given by:
n x nx P{D x} p (1 p) where x = 0, 1, 2, … x
If we took a large enough number of samples, we would find that the mean proportion defective in the distribution () would be very close to p, and that the population variance would be given by:
2 p(1 p) ˆ p n
If we wanted to do a two-sided hypothesis test to see if the proportion defective from one sample was different from the proportion defective found in another sample, we could use an approximate normal distribution and the test statistic:
pˆ pˆ z 1 2 0 n1pˆ 1 n2pˆ 2 1 1 where pˆ ˆ ˆ p(1 p) n1 n2 n1 n2
This hypothesis test lends itself to the creation of a control chart for the proportion defective if we are taking random samples from an industrial process. Like we did with the continuous variable control
08105240d77bcb3a747aa9a4451645f3.doc Page 2 of 8 Figure1.
Defective Units Is the size No, varies Use p-Chart of the (possibly with multiple inspection defects) sample Binomial Distribution fixed? Yes, Use np-Chart constant
Discret What is Individual Defects Is the size Yes, Use c-Chart e the of the constant inspection Poisson inspection Attribu basis? Distribution unit fixed? te
No, varies Use u-Chart
Kind of inspecti on variable ?
Continuou Range Which Use X-bar and s spread R-Chart method Variable preferred? Standard Deviation Use X-bar and S-Chart
08105240d77bcb3a747aa9a4451645f3.doc Page 3 of 8 charts, we’ll turn the hypothesis test on its’ side, and estimate a centerline and the upper and lower control limits.
The best guess for the unknown population proportion defective would be to find the mean proportion defective over a large number of samples, and let this become our centerline:
m m p D i i where m is the number of samples, each of size n p i1 i1 m mn
As before, when we do not yet have an idea of the process’ performance, we would estimate the control limits from a large number (20-25) of independent and random samples. Using plus and minus three standard deviations from the centerline (Shewhart style), the trial control limits for the proportion defective in any particular sample are:
p(1 p) UCL p 3 n CL p p(1 p) LCL p 3 n
These trial control limits would be plotted along with the individual, time-ordered sample data, and then checked to be sure that all samples were within the control limits. If not, we would investigate the out-of- control points, remove the special cause (if found) and recalculate the trial limits without any out-of-control samples in the data. Then we would use those control limits for production monitoring purposes.
If the control limits were to be calculated from a standard value (prior history) for the proportion defective, the formulation is similar (we replace the sample parameters with the standard population values):
p(1 p) UCL p 3 n CL p p(1 p) LCL p 3 n
If we desired control limits at a different point (either further out from, or closer into the mean) we could replace the constant 3 with a different value (2 for 2 limits, 6 for 6 limits…). Note also that we should pick our sample size so that there is a high probability of finding at least one defect in a sample – otherwise, we would effectively accept a zero-defects sample (rendering our lower control limit useless in detecting important shifts in our process).
In practice, however, the sample size for a p-chart does not have to be held constant. Usually, we would estimate the mean sample size ( n ) and substitute it for the fixed sample size (n) in the above equations. The computation for the mean sample size from m samples of differing sizes is found by:
m ni n i1 m
08105240d77bcb3a747aa9a4451645f3.doc Page 4 of 8 A more exact alternative would be to compute variable width control limits that change with the individual sample size. If we started with m samples (20 ≤ m ≤ 25) of individual size ni, then we would estimate the centerline (once) at p from:
m Di p i1 m ni i1
And then we could have the control limits vary in width about this centerline as the sample size changed, using the limits:
p(1 p) UCL p 3 ni CL p p(1 p) LCL p 3 ni
NOTE: When using variable width control limits, it is not possible to utilize rules for detecting runs. In general, run rules are never used with p-charts. The lack of a strong statistical basis for these run rules is one of the reasons that continuous variable control charts are preferred to attributes charts – there is simply more information available from the continuous variable than from the discrete variable.
NP-Charts
The np-Chart is used to track the number of defective units in a sample of units (rather than the proportion of defective units). Like the p-chart, this chart is derived from the Binomial Distribution. However, the np-Chart always requires a fixed sample size. Calculating the control limits from sample data leads to:
UCL np 3 np(1 p) CL np LCL np 3 np(1 p)
And if there was a historical standard for estimating np, then the control limits become:
UCL np 3 np(1 p) CL np LCL np 3 np(1 p)
For an np-Chart, the control limits are constant (until we improve the process and recalculate tighter control limits). In this case, as long as we have a sample of inspection units that has a high probability of having at least one defective unit in each sample, we can utilize the run rules without violating assumptions too much. This gives us a slightly more powerful control chart than the p-chart, at the cost of inspecting a slightly larger sample of the units.
08105240d77bcb3a747aa9a4451645f3.doc Page 5 of 8 C-Charts
Sometimes the presence of a defect does not “ruin” the product, even if defects are undesirable. For example, a farmer might still buy a tractor even with a few scratches in the paint on one fender, or a computer programmer might still accept an LCD monitor with one or two defective pixels. However, a good manufacturer would still wish to track the number of defects occurring in each product in order to improve and continue to compete. C- and u-Charts work to track the number of defects that occur as a product is created.
The c-chart is derived from the Poisson Distribution, which assumes that the opportunities for defects to occur is essentially infinite (ex.: small defects occurring within a large area). If x is a given number of defects, then the probability of observing x defects in an inspection unit is:
e c c x p(x) where c is the true mean count of the number of occurrences per unit x!
For the Poisson distribution, the mean and the variance are the same, and both are equal to c. This information can be used to set up an approximate Normal hypothesis test (but it is quicker to just cut to the derivation of the control charts limits!).
The mean count of defects occurring per inspection unit is best estimated by counting the total number of defects occurring over a large number of inspection units:
c total number of defects total inspection units
This parameter will represent our center line, but we will also need upper and lower control limits. If we are working to establish control limits from sample data, the formulation would be: UCL c 3 c CL c LCL c 3 c or 0 if LCL is negative
Alternatively, if we are continuing to use an existing and stable process, the “standard” value of c could be used for the control limits by:
UCL c 3 c CL c LCL c 3 c or 0 if LCL is negative
In all cases of the c-Chart, the inspection unit is a constant size. Provided that the LCL is greater than zero, then we will have constant control limits and we can apply the rules for detecting runs in addition to the out-of-control point criteria to determine if our process is stable and in-control.
U-Charts
These charts are used when the size of the inspection unit may vary. (In fact, the size might not even be an integer multiple of the inspection units!) Assuming that we have to generate the control limits from a pool of 20-25 samples, our best estimate for the center line for the u-chart is:
u total number of defects total units inspected
08105240d77bcb3a747aa9a4451645f3.doc Page 6 of 8 One option for the u-chart is to use the mean sample size in computing the upper and lower control limits. The mean sample size and the control limits are computed from:
m ni n i1 m
u UCL u 3 n CL u u LCL u 3 n
Another, more exact alternative is to use variable control limits. Similar to the variable limit p-chart, we would compute our centerline once from our sample data, and then use it to change the limits with each sample. From this point, we can compute our control limits for each individual sample size (ni) by:
u UCL u 3 ni CL u u LCL u 3 ni
As with the other variable limit control chart, the ability to use run tests is forfeited. Additionally, if the defects occur in clusters (ie. the presence of one defect makes it more likely for another defect to occur), then the defects do not follow a Poisson Distribution and the control limits will not be very precise. In some instances, mixtures of defect types can sometimes cause clustering.
In some cases, when the defect rates are in the low parts-per-million range, the size of the inspection unit will grow very large. U-Charts can also be used if the plotted variable is changed to be the time- between-successive-defects, with much lower inspection frequency/cost.
Summary of Continuous and Attribute Control Charts
In general, continuous variable control charts will detect smaller changes earlier than an attribute control charts can. The Central Limit Theorem can be used to justify an approximation of attribute data with control charts based on the Normal Distribution. Finally, continuous variable control charts normally require much smaller sample sizes as well.
However, attribute control charts can cover several defect types on one chart, where two charts (x-bar and R- or -Charts are required for each single characteristic to be measured. And continuous variables generally require more refined equipment and time to complete the measurement, leading to a higher inspection cost.
Assignment
Attribute control charts also lend themselves easily to “service” applications. Examples might include the number of incorrect invoices per customer, the proportion of incorrect orders taken in a day, the number of return service calls to resolve a problem, etc. Come up with at least one example of a non-
08105240d77bcb3a747aa9a4451645f3.doc Page 7 of 8 manufacturing application appropriate for each of the attributes control charts. Are there some that would be applicable for your engineering or technical department? How about those departments that support your own?
08105240d77bcb3a747aa9a4451645f3.doc Page 8 of 8