Total Quality Management Control Charts Excel Add-In Software

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Part 2. MM4XL Tools > 1. Strategic Tools > Quality Manager > 1. SPC Attribute Charts > P-Chart with Fix & Variable Lot

Quality Manager

P-Chart with Fix & Variable Lot

The P-chart is perhaps the most used control chart for attribute data. It can measure the number of nonconformities in lots of fixed or variable size. Nonconformities are results of a process that depart from normality. In the case of lots of fixed size, negative nonconformities could measure, for example, out of every 100 incoming calls the number waiting longer than 45 seconds for an operator to pick them. On the other hand, positive nonconformities could measure such things as the daily number of orders resulting from total cold calls.

The picture below shows a P-chart drawn with MM4XL's Quality Manager tool. Select one of two P Chart types. The variable lot type requires you to then select two variables for Num inspected and Num defective. The fixed lot type requires one variable only for Num defective. If an input range was not selected in window 1, the P-charts will not be available in the list of chart types and the right side of the window below will be blank. Click on Next to go to the window where you select options for printing the results on sheet.

Total Quality Management Control Charts Excel Add-In Software

Technical notes

The control concept of the P-chart is based on the Binomial probability distribution function. When the checkbox Simulate data in the window above is checked, Quality Manager shows a series of Binomial random numbers (thin green line) before with the user input data (thick blue line). The simulated data help you to understand whether the process is following a stable pattern or not. Unstable processes need to be stabilized in order to be correctly analyzed with control charts. If the shape of the user data is remarkably different from that of the simulated data, one can reasonably conclude that the user data could be influenced by some kind of external force. That is, the impact on the input data should be removed and a new analysis should be run.

In order to detect a change, the input data is shown in a chart within boundaries as in the following picture. The control limits are placed at 3 standard deviations (see field Z in the window) above and below the average (Pbar) of nonconformities. Measurements falling outside control limits indicate a change in the process.

The limits for the P-chart with fixed lot are set as follows:

The limits for the P-chart with variable lot are set as follows:

Using P-charts to detect change in a process is like saying that as long as results lie within the three standard deviations from the mean the process is seen as working correctly. To calibrate the chart it is advisable to work with data series comprising 20 to 40 base measurements. Too short a series may depart seriously from the shape of the Binomial distribution and, therefore, produce unreliable control charts.

When an item goes beyond control limits the chart has identified a change. The change can be bad or good according to the measurement data. Returning to the opening examples, an increase in calls waiting longer than 45 seconds to be picked up by an operator is a bad change, and the process needs to be adjusted to continue working as planned. On the other hand, an increase in orders on cold calls is a good change, and the source of variation should be clearly identified in order to make the change common practice in the process. Items beyond limits are highlighted with a red, round marker, as shown in the window above.

Input data for the fixed lot

The input data for the P-chart with fixed lot requires only one column of counts. The picture below shows a suitable data series in the range A1:A35, mind the hidden rows. These can be negative or positive nonconformities.

Output results for the fixed lot

The P-chart with fixed lot can show in output two charts and three tables according to the user selection in the third window (see section Introduction to Quality Manager). The first table, in the picture below, contains indexes that describe the input and simulated data in terms of:

Size of the variable: Max, Min, Sum, Range and Counts
Central tendency: Average, Median, Mode and Standard deviation (of a variable)
Chart limits: Upper Control Limit (UCL), Pbar, Lower Control Limit (LCL)
Z stands for the number of standard deviations where the control limits should be placed
Sigma is the standard deviation of a subgroup

For the sake of brevity, the second table is not shown here. In 6 columns it shows the details of the chart limits by item. In the picture below, the small, red triangle in the upper right corner of the first column label is a Comment that displays a short message. A number of comments are created by Quality Manager. Place the mouse pointer on the red triangle to display the message. The column Fr/def stands for the fraction of defectives.

The P-Chart with fixed lot size in the picture below refers to an input variable (thick blue line) without observations outside of control limits. The thin green line on the left side refers to simulated random data produced by Quality Manager, in this example, according to the Binomial distribution. Comparing the random data to the user input data can help you get a visual understanding of the departure of the input data from normality. In this case, both simulated and user data take a shape that does not indicate any particular sign of an existing trend. Therefore, we could conclude that the input data is stable and can be used for the purpose of control. For a better way to assess normality, read the material on the Process Capability tool available in Quality Manager.

Input data for the variable lot

The input data for the P-chart with variable lot requires two columns of counts. The picture below shows a suitable data series in the range B6:C31, mind the hidden rows. These can be negative or positive nonconformities.

Output results for the variable lot

The printout of the P-chart with variable lot size looks exactly like that of the P-chart with fixed lot with the exception that the chart (see picture below) shows variant control limits (dotted lines) rather than straight lines.

The histogram in the picture below shows two series:

The blue bars refer to the observed frequency of count classes in the input data. The first bar, for instance, tells us that there is 1 count for items equal to or smaller than 25 in the data. The second bar shows 3 counts for items larger than 25 and smaller than 52.8. And so on for all bars.
The bell-shaped red line shows the expected normal curve for a variable with the same range as the input data, and it helps to verify with a quick visual inspection whether the input data follow a normal distribution or not. P-charts, however, follow the Binomial distribution that they tend to approximate normality only with a large number of observations.

Tip: When working with P and NP charts, sometimes it is necessary to adjust the Pbar value in order to align the central line (Pbar) of simulated and user data. When the two lines lie roughly at the same level one can safely assume the simulated data reflect the shape of the data input by the user. For P charts, the P-bar value is shown in the lower right area of the window among the statistics. An estimate of the NP-bar can be found by first running a P chart with fixed lot and then applying the P-bar to the NP chart.

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