Part 2. MM4XL Tools > 1. Strategic Tools > Quality Manager > 1. SPC Attribute Charts > C-chart

Quality Manager

C-chart

The C-chart measures the number of nonconformities in a single item. Nonconformities are results of a process that depart from normality. For instance, newspaper pages are supposed to be free of errors, so errors (defects) are the nonconformities in the page. The number of customers receiving wrong items in their order are also nonconformities, as are the number of daily purchases with a coupon in a supermarket or any other data series measuring incompliance with counts at different moments in time.

User selections

The picture below shows a C-chart drawn with MM4XL's Quality Manager tool. After selecting Chart type, as shown in the following picture, if you have selected a range with more than one variable (column) in input, choose the variable for Num defectives to analyze, otherwise, the tool will show automatically the data of the only input series available. If an input range was not selected in window 1, the C-chart 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 C-chart is based on a low presence of nonconformities with a high probability of occurrence, and such processes behave according to the Poisson probability distribution function. When the checkbox Simulate data in the window above is checked, Quality Manager shows a series of Poisson random numbers (thin green line) before the user input data (thick blue line). The simulated data helps 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 may 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.

Total Quality Management Control Charts Excel Add-In Software


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 second window) above and below the average (Cbar) of nonconformities. Measurements falling outside control limits indicate a change in the process.

Total Quality Management Control Charts Excel Add-In Software


Using the C-chart to detect change in a process is like saying that as long as results lie within the 3 standard deviations from the mean the process is seen as working correctly. In this situation it is advisable to work with data series comprising 20 to 40 base measurements useful to calibrate the chart. A too-short series may depart seriously from the shape of the Poisson distribution and, therefore, produce an unreliable control chart.

The LCL cannot go below the zero. When an item goes beyond the UCL the chart has found a change. The change can be bad or good according to the measurement data. If the data refer to, for instance, errors in the orders delivered to clients, the change is bad and the source of change should be identified and removed from the process. If the data refer, for instance, to orders placed in a direct marketing campaign, the change is good and the source producing the change should be made common practice in the process. Items beyond limits are highlighted with a red, round marker, as shown in the window on the previous page.

Input data

The input data for the C-chart requires one single column of counts. The picture below shows a suitable data series in the range A1:A51, mind the hidden rows. These can be negative nonconformities, such as defects occurring every hour, or they can be positive nonconformities, such as daily sales from a direct marketing campaign.

Total Quality Management Control Charts Excel Add-In Software

Tip:
In order to speed up the tool, uncheck the Simulate data option when working with long data series.



Output results

The C-chart 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, shown below, contains indexes describing the input 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), Cbar, Lower Control Limit (LCL), Sigma
  • Z stands for the number of standard deviations where the control limits should be placed
  • Sigma is the standard deviation of a subgroup

Total Quality Management Control Charts Excel Add-In Software


For the sake of brevity, the second table is not shown here. In 5 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 which 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.

Total Quality Management Control Charts Excel Add-In Software


The C Control Chart in the picture below refers to an input variable (thick blue line) presenting one observation outside the UCL while all other points lie within limits. The thin green line on the left refers to simulated random data that Quality Manager produced, in this case, according to the Poisson 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 show any particular sign of an existing trend. Therefore, we could believe the input data is stable and can be used for the purpose of control. The peak outside of the limit may be due to random variation. Should the peak exceed the control limit remarkably, you need to explain the reasons for departure and try to stabilize the process, removing the noise. For a better way to assess normality, read also the material on the Process Capability tool available in Quality Manager.

Total Quality Management Control Charts Excel Add-In Software


The third table is made of four columns relating to the data for the histogram chart. The Levels column refers to the intervals classes. n shows the counts for each class, and Count Exp reports the expected number of items in each class for a normally distributed variable.

Total Quality Management Control Charts Excel Add-In Software


The histogram in the picture below shows two series:

  • The blue bars refer to the observed frequency of count classes in the input data, and come from column n. The first bar, for instance, tells us that there are 25 zeros, or sampled items without nonconformities. The second bar shows 16 counts for ones in the data (although the axis value below the second bar is 1.3 due to a rounding effect). 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. It is created with the values from column Count Exp, and it helps to verify with a quick visual inspection whether the input data follow a normal distribution or not. C-charts, however, follow the Poisson distribution that they tend to approximate normality only with a large number of observations.

Total Quality Management Control Charts Excel Add-In Software

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