9. p Charts
The x bar R chart plots variable data. Other chart types, such as the p chart, plot attribute data. Examples are go/no-go or accept/reject rates.
The p chart is an attribute chart. The p chart usually plots the proportion of non-conforming items.

To distinguish failing items from passing items, the standards for acceptance and rejection must be clearly defined. Since the failure rate is usually small, you need to take large samples– 50, 100, or more, to avoid a lot of zero data entries. The number of defectives should be 5 or more per sample in order to use the methods of this section (The methods are based on a normal distribution of "successes", in this case finding a non-conforming item.) Also, the sample size should be consistent, within 25% of the average sample size. Since there is only one value plotted (p) there is no range portion to chart.

Calculating p examples
One hundred moldings are inspected; 2 have cavities, 7 do not pass the warp gauge, 0 have flash.
p = = 0.09, the proportion of nonconforming in all three categories combined.
Another example: In 500 items, 16 are nonconforming.
The proportion non-conforming is = 0.032
p chart procedure

1. Calculate p for each sample
2. Set up the control chart scale
3. Plot the p values, at least 20 points to establish a history.
4. Calculate , the average p value, and the control limits.
5. The interpretation and maintenance of the p chart is similar to the x-bar R chart.
p Chart Statistics

n = the sample size, the number of individuals in a sample.
k = the number of samples.
N = number of individuals, the total number of observations. N = n times k if all samples are the same size.
p = the fraction of "successes", which may be defective items, expressed as a decimal.
1–p = the fraction of "failures", which may be passing items. Sometimes represented by q.
p+q = 1, so q = 1–p. For example, if the proportion of non-conforming items is p = 0.15, the proportion of
conforming items is 1– 0.15 = 0.85. So 1 – p = q = 0.85
= average p of all samples. This is the same as the average of the p's of all the individual samples, .

When the sample sizes are not constant use the average sample size, , for n in control limit calculations:

=
Notice that 3 standard deviations is estimated by . One standard deviation is .
p chart calculation example

This is a very small sample so that the calculation is more understandable.
We take three samples of n = 100 each. The results are

sample number
n
 number nonconforming
p
1–p = q
1
100
9
.09
.91
2
100
12
.12
.88
3
100
6
.06
.94
averages
100
9
.09 =
.91 =
UCLp = = .119

LCLp = = .061

#38 Practice on p chart Statistics
A personnel section inspects intake records for blank entries, going back over five months. A record is deemed nonconforming if it has one or more blanks. In 5 samples of 100 records each, the following number of records are nonconforming: 14, 12, 5, 7, 10.

N = ___________ k = ____________ n = ____________ = ___________

The p's are _____, _____, _____, _____, _____. = ___________ 1- = ___________

The standard deviation is ________________.

UCLp = __________

LCLp = __________
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