Question

p Chart A variation of the control chart for p is the np chart, in which the actual numbers of defects are plotted instead of the proportions of defects. The np chart has a centerline value of \(n\bar p\), and the control limits have values of \(n\bar p + 3\sqrt {n\bar p\bar q} \)and\(n\bar p - 3\sqrt {n\bar p\bar q} \). The p chart and the np chart differ only in the scale of values used for the vertical axis. Construct the np chart for Example 1 “Defective Aircraft Altimeters” in this section. Compare the np chart to the control chart for p given in this section

Short answer

The np chart constructed is shown below:

From the constructed np chart, the process is not within statistical control because

• there is at least one point beyond the upper control limit, and
• there seems to be an upward trend in the number of defects.

For the given data, the p chart and the np chart are nearly identical and have the same structure. The only variation between the two graphs is in the vertical scale values.

Step-by-step solution

Given information

Data are given on the number of defective altimeters in 12 samples.

The size of each sample is 100.

np Chart

The np chart is an attribute control chart that depicts the number of defects in the individual samples as compared to the p chart, which shows the proportion of defects in each sample.

The following data is utilized to construct the np chart, which shows the number of defective altimeters in each sample:

Sample number	Number of defects
1	2
2	0
3	1
4	3
5	1
6	2
7	2
8	4
9	3
10	5
11	12
12	7

Important values of thenp chart

Let\(n\bar p\)be the estimated number of defectivealtimetersin all the samples.

It is computed as follows:

\(\begin{array}{c}n\bar p = n \times \left( {\frac{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{defectives}}\;{\rm{from}}\;{\rm{all}}\;{\rm{samples}}\;{\rm{combined}}}}{{{\rm{Total}}\;{\rm{number}}\;{\rm{of}}\;{\rm{samples}}}}} \right)\\ = \left( {100} \right) \times \left( {\frac{{2 + 0 + 1 + ..... + 7}}{{12\left( {100} \right)}}} \right)\\ = \left( {100} \right) \times \left( {\frac{{42}}{{1200}}} \right)\\ = \left( {100} \right) \times \left( {0.035} \right)\end{array}\)

\( = 3.5\)

The value of\(\bar q\)is computed as shown:

\(\begin{array}{c}\bar q = 1 - \bar p\\ = 1 - 0.035\\ = 0.965\end{array}\)

The value ofthe central line is calculated below:

\(\begin{array}{c}CL = n\bar p\\ = 3.5\end{array}\)

The lower control limit (LCL) is computed below:

\(\begin{array}{c}LCL = n\bar p - 3\sqrt {n\bar p\bar q} \\ = 3.5 - 3\sqrt {\left( {100} \right)\left( {0.035} \right)\left( {0.965} \right)} \\ = - 2.01339\\ \approx 0\end{array}\)

The upper control limit (UCL) is computed below:

\(\begin{array}{c}UCL = n\bar p + 3\sqrt {n\bar p\bar q} \\ = 3.5 + 3\sqrt {\left( {100} \right)\left( {0.035} \right)\left( {0.965} \right)} \\ = 9.01\end{array}\)

Tabulation of the number of defectives

The following table shows the number of defective altimeters corresponding to the sample number:

Serial number	Number of defectives (d)
1	2
2	0
3	1
4	3
5	1
6	2
7	2
8	4
9	3
10	5
11	12
12	7

Construction

Follow the given steps to construct the p chart:

• Mark the values 1, 2, 3 ...,12 on the horizontal axis and label it “Sample.”
• Mark the values 0, 2, 4 ...,14 on the vertical axis and label it “Sample Count.”
• Plot a horizontal line parallel to the horizontal axis corresponding to the value “3.5” on the vertical axis and label the line (on the left side) “\(n\bar p\)= 3.5.”
• Plot a horizontal line parallel to the horizontal axis corresponding to the value “9.01” on the vertical axis and label the line (on the left side) “UCL= 9.01.”
• Plot a horizontal line parallel to the horizontal axis corresponding to the value “0” on the vertical axis and label the line (on the left side) “LCL= 0.”
• Mark the 12 samplepoints on the graph and join the dots using straight lines.

The following np chart is obtained:

Analysis of the np chart

The following characteristics can be observed from the plotted chart:

• There is at least one point beyond the upper control limit.
• There appears to be an upward trend in the number of defects.

Since the above criteria point towards the violation of the stability of the given process, the process is not under statistical control.

Comparison of np chart and p chart

Referring to Example 1, the centerline and control limits (LCL and UCL) are given as follows:

\(\begin{array}{c}CL = 0.035\\UCL = 0.090\\LCL = - 0.020\\ \approx 0\end{array}\)

The p chart and the np chart plotted for the given data are approximately identical and have the same structure. The only difference in the two charts is in the values marked on the vertical scale.