Chapter 4, Problem 4.69P

To determine

The control limits.

Answer to Problem 4.69P

The Upper control limit for averages, lower control limit for averages, upper control limit for ranges and lower control limit for ranges is $0.72433$ , $0.62227$ , $0.15974$ and $0$ .

Explanation of Solution

Given:

The number of sample subsets are $6$ .

The number of measured values in each subset are $4$ .

Formula used:

The expression for average of sample subset is given as,$\overline{x}=\frac{x_1+x_2+x_3+x_4}{4}$

Here, $x_1$ , $x_2$ , $x_3$ and $x_4$ are the variables of subset.

The expression forrange of sample subset is given as,$R=x_{\max }-x_{\min }$

Here, $x_{\max }$ is the maximum value in subset and $x_{\min }$ is the minimum value in subset.

The expression for average of averages is given as,$\overline{\overline{x}}=\frac{{\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5+{\overline{x}}_6}{6}$

The expression for average of ranges is given as,$\overline{R}=\frac{R_1+R_2+R_3+R_4+R_5+R_6}{6}$

The expression for upper control limit for the averages is given as,$UC{L}_x=\overline{\overline{x}}+A_2\overline{R}$

Here, $A_2$ is constant.

The expression for lower control limit for the averages is given as,$LC{L}_x=\overline{\overline{x}}-A_2\overline{R}$

The expression for upper control limit for the ranges is given as,

  $UC{L}_R=D_4\overline{R}$

Here, $D_4$ is the constant.

The expression for lower control limit for the ranges is given as,

  $LC{L}_R=D_3\overline{R}$

Here, $D_3$ is the constant.

Calculation:

Refer to table 4.2 “Constant for control charts” for sample size 4 values obtained,

  $\begin{array}{l}{A}_2=0.729\\ D_4=2.282\\ D_3=0\end{array}$

The average of the first subset can be calculated as,

  $\begin{array}{l}{\overline{x}}_1=\frac{0.65+0.75+0.67+0.65}{4}\\ {\overline{x}}_1=\frac{2.72}{4}\\ {\overline{x}}_1=0.68\end{array}$

The range of the first subset can be calculated as,

  $\begin{array}{l}{R}_1=0.75-0.65\\ R_1=0.10\end{array}$

The average of the second subset can be calculated as,

  $\begin{array}{l}{\overline{x}}_2=\frac{0.69+0.73+0.70+0.68}{4}\\ {\overline{x}}_2=\frac{2.8}{4}\\ {\overline{x}}_2=0.7\end{array}$

The range of the second subset can be calculated as,

  $\begin{array}{l}{R}_2=0.73-0.68\\ R_2=0.05\end{array}$

The average of the third subset can be calculated as,

  $\begin{array}{l}{\overline{x}}_3=\frac{0.65+0.68+0.65+0.61}{4}\\ {\overline{x}}_3=\frac{2.59}{4}\\ {\overline{x}}_3=0.6475\end{array}$

The range of the third subset can be calculated as,

  $\begin{array}{l}{R}_3=0.68-0.61\\ R_3=0.07\end{array}$

The average of the fourth subset can be calculated as,

  $\begin{array}{l}{\overline{x}}_4=\frac{0.64+0.65+0.60+0.60}{4}\\ {\overline{x}}_4=\frac{2.49}{4}\\ {\overline{x}}_4=0.6225\end{array}$

The range of the fourth subset can be calculated as,

  $\begin{array}{l}{R}_4=0.65-0.60\\ R_4=0.05\end{array}$

The average of the fifth subset can be calculated as,

  $\begin{array}{l}{\overline{x}}_5=\frac{0.68+0.72+0.70+0.66}{4}\\ {\overline{x}}_5=\frac{2.76}{4}\\ {\overline{x}}_5=0.69\end{array}$

The range of the fifth subset can be calculated as,

  $\begin{array}{l}{R}_5=0.72-0.66\\ R_5=0.06\end{array}$

The average of the sixth subset can be calculated as,

  $\begin{array}{l}{\overline{x}}_6=\frac{0.70+0.74+0.65+0.71}{4}\\ {\overline{x}}_6=\frac{2.8}{4}\\ {\overline{x}}_6=0.7\end{array}$

The range of the sixth subset can be calculated as,

  $\begin{array}{l}{R}_6=0.74-0.65\\ R_6=0.09\end{array}$

The average of averages can be calculated as,

  $\begin{array}{l}\overline{\overline{x}}=\frac{{\overline{x}}_1+{\overline{x}}_2+{\overline{x}}_3+{\overline{x}}_4+{\overline{x}}_5+{\overline{x}}_6}{6}\\ \overline{\overline{x}}=\frac{0.68+0.7+0.6475+0.6225+0.69+0.7}{6}\\ \overline{\overline{x}}=\frac{4.04}{6}\\ \overline{\overline{x}}=0.6733\end{array}$

The average of ranges can be calculated as,

  $\begin{array}{l}\overline{R}=\frac{R_1+R_2+R_3+R_4+R_5+R_6}{6}\\ \overline{R}=\frac{0.10+0.05+0.07+0.05+0.06+0.09}{6}\\ \overline{R}=\frac{0.42}{6}\\ \overline{R}=0.07\end{array}$

The upper control limit for the averages can be calculated as,

  $\begin{array}{l}UC{L}_x=\overline{\overline{x}}+A_2\overline{R}\\ UC{L}_x=0.6733+\left(0.729\times 0.07\right)\\ UC{L}_x=0.72433\end{array}$

The lower control limit for the averages can be calculated as,

  $\begin{array}{l}LC{L}_x=\overline{\overline{x}}-A_2\overline{R}\\ LC{L}_x=0.6733-\left(0.729\times 0.07\right)\\ LC{L}_x=0.62227\end{array}$

The upper control limit for the ranges can be calculated as,

  $\begin{array}{l}UC{L}_R=D_4\overline{R}\\ UC{L}_R=2.282\times 0.07\\ UC{L}_R=0.15974\end{array}$

The lower control limit for the ranges can be calculated as,

  $\begin{array}{l}LC{L}_R=D_3\overline{R}\\ LC{L}_R=0\times 0.07\\ LC{L}_R=0\end{array}$

Conclusion:

Therefore, the Upper control limit for averages, lower control limit for averages, upper control limit for ranges and lower control limit for ranges is $0.72433$ , $0.62227$ , $0.15974$ and $0$ .