# Statistical Process Control

## Example 1

Assume that in the manufacture of 1 kg Mischmetal ingots, the product weight varies with the batch. Below are a number of subsets taken at normal operating conditions (subsets 1-7), with the weight values given in kg. Construct the $X$-Bar, $R$-charts, and $S$-charts for the experimental data (subsets 8-11). Measurements are taken sequentially in increasing subset number.

 Subset # Values (kg) 1 (control) 1.02, 1.03, 0.98, 0.99 2 (control) 0.96, 1.01, 1.02, 1.01 3 (control) 0.99, 1.02, 1.03, 0.98 4 (control) 0.96, 0.97, 1.02, 0.98 5 (control) 1.03, 1.04, 0.95, 1.00 6 (control) 0.99, 0.99, 1.00, 0.97 7 (control) 1.02, 0.98, 1.01, 1.02 8 (experimental) 1.02, 0.99, 1.01, 0.99 9 (experimental) 1.01, 0.99, 0.97, 1.03 10 (experimental) 1.02, 0.98, 0.99, 1.00 11 (experimental) 0.98, 0.97, 1.02, 1.03

Solution:

First, the average, range, and standard deviation are calculated for each subset.

 Subset # Values (kg) Average (cc) Range (R) Standard Deviation(s) 1 (control) 1.02, 1.03, 0.98, 0.99 1.0050 0.05 0.023805 2 (control) 0.96, 1.01, 1.02, 1.01 1.0000 0.06 0.027080 3 (control) 0.99, 1.02, 1.03, 0.98 1.0050 0.05 0.023806 4 (control) 0.96, 0.97, 1.02, 0.98 0.9825 0.06 0.026300 5 (control) 1.03, 1.04, 0.95, 1.00 1.0150 0.09 0.040509 6 (control) 0.99, 0.99, 1.00, 0.97 0.9875 0.03 0.022583 7 (control) 1.02, 0.98, 1.01, 1.02 1.0075 0.04 0.028930 8 (experimental) 1.02, 0.99, 1.01, 0.99 1.0025 0.03 0.025000 9 (experimental) 1.01, 0.99, 0.97, 1.03 1.0000 0.06 0.025820 10 (experimental) 1.02, 0.98, 0.99, 1.00 0.9975 0.04 0.027078 11 (experimental) 0.98, 0.97, 1.02, 1.03 1.0000 0.06 0.029409

Next, the grand average $X_{G A}$, average range $R_{A}$, and average standard deviation $S_{A}$ are computed for the subsets taken under normal operating conditions, and thus the centerlines are known. Here $n=4$.

\begin{aligned} &X_{G A}=1.0004 \\ &R_{A}=0.05428 \\ &S_{A}=0.023948 \end{aligned}

$X$-Bar limits are computed (using $R_{A}$).

\begin{aligned} &\mathrm{UCL}=X_{G A}+A_{2} R_{A}=1.0004+0.729(0.05428)=1.04 \\ &\mathrm{LCL}=X_{G A}-A_{2} R_{A}=1.0004-0.729(0.05428)=0.96 \end{aligned}

$X$-Bar limits are computed (using $S_{A}$).

\begin{aligned} &\mathrm{UCL}=X_{G A}+A_{3} S_{A}=1.0004+1.628(0.023948)=1.04 \\ &\mathrm{LCL}=X_{G A}-A_{3} S_{A}=1.0004-1.628(0.023948)=0.96 \end{aligned}

Note: Since $n=4$ (a relatively small subset size), both $R_{A}$ and $S_{A}$ can be used to accurately calculate the UCL and LCL.

$R$-chart limits are computed.

$\begin{gathered} \mathrm{UCL}=D_{4} R_{A}=2.282(0.05428)=0.12 \\ \mathrm{LCL}=D_{3} R_{A}=0(0.05428)=0 \end{gathered}$

$S$-chart limits are computed.

$\begin{gathered} \mathrm{UCL}=B_{4} S_{A}=2.266(0.023948)=0.054266 \\ \mathrm{LCL}=B_{3} S_{A}=0(0.023948)=0 \end{gathered}$

The individual points in subsets 8-11 are plotted below to demonstrate how they vary with in comparison with the control limits.

Figure E-1: Chart of individual points in subsets 8-11.

The subgroup averages are shown in the following $X$-Bar chart:

Figure E-2: $X$-Bar chart for subsets 8-11.

The $R$-chart is shown below:

Figure E-3: $R$-chart for subsets 8-11.

The $S$-chart is shown below:

Figure E-4: $S$-chart for subsets 8-11.

The experimental data is shown to be in control, as it obeys all of the rules given above.