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?
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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:
p̂ 
D
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
P{D  x }   p x (1  p) n x
x
where x = 0, 1, 2, …
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:
 p̂2 
p(1  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:
z0 
p̂1  p̂ 2
 1
1
p̂(1  p̂)

 n1 n 2



where
p̂ 
n1p̂ 1  n 2 p̂ 2
n1  n 2
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
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Figure1.
Defective Units
(possibly with multiple defects)
Binomial Distribution
Is the size of
the inspection
sample fixed?
No, varies
Yes, constant
Discrete
Attribute
Individual Defects
What is the
inspection
basis?
Poisson Distribution
Is the size of
the inspection
unit fixed?
Yes, constant
No, varies
Use p-Chart
Use np-Chart
Use c-Chart
Use u-Chart
Kind of
inspection
variable?
Continuous
Variable
Which spread
method
preferred?
Range
Use X-bar and
R-Chart
Standard Deviation
Use X-bar and
S-Chart
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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
p
i1

m
i
i1
mn
where m is the number of samples, each of size n
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:
UCL  p  3
p(1  p)
n
CL  p
LCL  p  3
p(1  p)
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-ofcontrol 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):
UCL  p  3
p(1  p)
n
CL  p
LCL  p  3
p(1  p)
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
n
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n
i
i1
m
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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
p
D
i
i1
m
n
i
i1
And then we could have the control limits vary in width about this centerline as the sample size changed,
using the limits:
UCL  p  3
p(1  p)
ni
CL  p
LCL  p  3
p(1  p)
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.
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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:
p( x) 
e c c x
x!
where c is the true mean count of the number of occurrences per unit
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
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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
n
n
i
i1
m
UCL  u  3
u
n
CL  u
LCL  u  3
u
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:
UCL  u  3
u
ni
CL  u
LCL  u  3
u
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 timebetween-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-
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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?
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