## Statistical Process Control

Read this chapter on the basics of statistical process control (SPC). SPC is a standard tool for monitoring whether a process is performing as expected and, if not, where problems occur. While reading, consider how this kind of tool factors in process capacity management.

### Example 2

It’s your first day on the job as a chemical engineer in a plant, and one of your responsibilities is to monitor the pH of a particular process. You are asked by your boss to monitor the stability of the system. She gives you some baseline data for the process, and you collect data for the process during your first day. Construct $X$-bar and $R$-Charts to report your results.

Table 1: Baseline data

 time (hours) ph 1 7.00 7.30 6.99 7.00 2 7.12 7.25 7.12 7.20 3 7.20 7.16 7.20 7.16 4 6.98 7.00 6.94 7.00 5 6.99 6.99 6.99 6.98 6 7.00 6.93 7.02 6.93 7 6.92 7.00 6.92 7.02 8 6.88 6.82 6.94 6.99 9 7.10 7.00 7.00 7.00 10 7.21 7.02 7.21 7.04 11 7.01 6.86 7.01 6.90 12 6.86 6.98 6.90 6.98 13 6.90 7.00 6.87 7.00 14 7.01 7.04 7.01 7.05 15 7.00 6.95 7.00 6.99 16 7.09 7.20 7.03 7.20 17 6.89 7.14 6.87 7.15 18 6.98 6.80 6.98 6.89 19 7.00 6.90 7.00 6.90 20 7.20 7.00 7.23 7.00 21 7.04 7.03 7.08 7.00 22 6.90 6.92 6.98 6.92 23 7.00 7.00 7.00 7.00 24 7.00 6.97 7.01 6.98

To be consistent with the baseline data, each hour you take four pH readings. The data you collect is displayed below.

Table 2: Experimental data

 time (hours) pH pH pH pH 1 6.99 6.99 7.00 6.89 2 6.98 7.12 7.05 6.96 3 7.00 7.18 7.08 7.04 4 7.01 6.94 6.98 7.00 5 6.90 6.99 6.93 7.01 6 6.96 7.01 7.00 7.14 7 7.04 6.92 6.82 7.01 8 7.00 6.93 7.00 6.90 9 7.01 7.00 7.02 6.92 10 7.04 7.18 6.99 6.93 11 6.91 7.01 6.90 7.00 12 7.00 6.97 6.98 7.18 13 7.00 6.89 7.00 7.03 14 7.03 7.01 7.05 6.87 15 6.97 7.00 7.00 6.98 16 7.03 6.97 7.02 6.98 17 6.99 6.89 6.87 6.99 18 6.89 6.98 6.98 6.98 19 6.98 7.00 7.00 7.02 20 7.02 7.15 6.97 6.98 21 7.02 7.08 7.08 7.00 22 6.97 7.01 6.98 7.05 23 7.01 7.04 6.99 7.08 24 6.97 7.00 6.98 6.98

Solution

For this situation, there are $\mathrm{k}=24$ subsets because there are 24 data sets. For each subset, $\mathrm{n}=4$ because there are four pH measurements taken each hour. The first thing you do is calculate the mean and range of each subset. The means are calculated using the AVERAGE() Excel function and the ranges are calculated using MAX() - MIN(). Once these values are calculated, the Grand Average $X_{G A}$ and average range $R_{A}$ are calculated. These values are simply the means of each subset's mean and range. This data is displayed below.

Table 3: Data used to calculate and grand average and Range.

Control Data

 time (hours) pH x_ave range 1 7.00 7.30 6.99 7.00 7.07 0.31 2 7.12 7.25 7.12 7.20 7.17 0.13 3 7.20 7.16 7.20 7.16 7.18 0.04 4 6.98 7.00 6.94 7.00 6.98 0.06 5 6.99 6.99 6.99 6.98 6.99 0.01 6 7.00 6.93 7.02 6.93 6.97 0.09 7 6.92 7.00 6.92 7.02 6.97 0.10 8 6.88 6.82 6.94 6.99 6.91 0.17 9 7.10 7.00 7.00 7.00 7.03 0.10 10 7.21 7.02 7.21 7.04 7.12 0.19 11 7.01 6.86 7.01 6.90 6.95 0.15 12 6.86 6.98 6.90 6.98 6.93 0.12 13 6.90 7.00 6.87 7.00 6.94 0.13 14 7.01 7.04 7.01 7.05 7.03 0.04 15 7.00 6.95 7.00 6.99 6.99 0.05 16 7.09 7.20 7.03 7.20 7.13 0.17 17 6.89 7.14 6.87 7.15 7.01 0.28 18 6.98 6.80 6.98 6.89 6.91 0.18 19 7.00 6.90 7.00 6.90 6.95 0.10 20 7.20 7.00 7.23 7.00 7.11 0.23 21 7.04 7.03 7.08 7.00 7.04 0.08 22 6.90 6.92 6.98 6.92 6.93 0.08 23 7.00 7.00 7.00 7.00 7.00 0.00 24 7.00 6.97 7.01 6.98 6.99 0.04 Means: 7.01 0.12

Now that you know $X_{G A}=7.01$ and $R_{A}=0.12$, you can calculate the upper control limit, $U C L_{t}$ and lower control limit, $L C L$, for the $X$-bar control chart.

From Table $A, A_{2}=0.729$ when $n=4$. Using equations UCL and LCL for $X$-bar charts listed above:

\begin{aligned}&\mathrm{UCL}=7.01+0.729(0.12)=7.0982 \\&\mathrm{LCL}=7.01-0.729(0.12)=6.9251\end{aligned}

Then the $U C L=7.0982, L C L=6.9251$ and $X_{G A}=7.01$ are plotted in Excel along with the average values of each subset from the experimental data to produce the $X$-bar control chart.

 time (hours) pH x_ave range 1 6.99 6.99 7.00 6.89 6.97 0.11 2 6.98 7.12 7.05 6.96 7.03 0.16 3 7.00 7.18 7.08 7.04 7.08 0.18 4 7.01 6.94 6.98 7.00 6.98 0.07 5 6.90 6.99 6.93 7.01 6.96 0.11 6 7.96 7.01 7.00 7.14 7.03 0.18 7 7.04 6.92 6.82 7.01 6.95 0.22 8 7.00 6.93 7.00 6.90 6.96 0.10 9 7.01 7.18 7.02 6.92 6.99 0.10 10 7.04 7.01 6.99 6.93 7.04 0.25 11 6.91 6.86 6.90 7.00 6.96 0.11 12 7.00 6.97 6.98 7.18 7.03 0.21 13 7.00 6.89 7.00 7.03 6.98 0.14 14 7.03 7.01 7.05 6.87 6.99 0.18 15 6.97 7.00 7.00 6.98 6.99 0.03 16 7.03 6.97 7.02 6.98 7.00 0.06 17 6.99 6.89 6.87 6.99 6.94 0.12 18 6.89 6.98 6.98 6.98 6.96 0.09 19 6.98 7.00 7.00 7.02 7.00 0.04 20 7.02 7.15 6.97 6.98 7.03 0.18 21 7.02 7.08 7.08 7.00 7.05 0.08 22 6.97 7.01 6.98 7.05 7.00 0.08 23 7.01 7.04 6.99 7.08 7.03 0.09 24 6.97 7.00 6.98 6.98 6.98 0.04

Table 4: Average subset values and ranges plotted on the $X$-bar and $R$-chart

Then, to construct the Range charts, the upper and lower control limits were found. For $\mathrm{n}=4, D_{3}=0$ and $D_{4}=2.282$ so then:

$\begin{gathered}\mathrm{LCL}=D_{3} R_{A}=0(0.12)=0 \\\mathrm{UCL}=D_{4} R_{A}=2.282(0.12)=0.2710\end{gathered}$

From both of these charts, the process is in control because all rules for stabilty are met.

Rule 1: No point falls beyond the UCl and LCL.

Rule 2: Two out of three consecutive points do not fall beyond $2\sigma$ on the same side of the centerline.

Rule 3: Four out of five consecutive points do not fall beyond $1\sigma$ on the same side of the centerline.

Rule 4: Nine or more consecutive points do not fall on the same side of the centerline.

It's important that both of these charts be used for a given set of data because it is possible that a point could be beyond the control band in the Range chart while nothing is out of control on the $X$-bar chart.

Another issue worth noting is that if the control charts for this pH data did show some points beyond the LCL or UCL, this does not necessarily mean that the process itself is out of control. It probably just means that the pH sensor needs to be recalibrated.