Chapter 16, Problem 2CSQD

Summary Introduction

Interpretation: The control over production process is to be assessed using the $\overline{\text{x}}$ and R charts.

Concept Introduction: $\overline{\text{x}}$ and R charts are tools of statistical process control used to monitor the impact of process improvement theories. Data collected as subgroups over a period of time and used for analysis.

Answer to Problem 2CSQD

The mean and range values are well within the control limits as seen from the x bar and R charts. Hence, the production process is in control.

Explanation of Solution

Given information:

The $\overline{\text{x}}$ and R charts are plotted to visualize the process means and dispersion of the data distribution. The factors from the standard 3 sigma chart are used to compute the control limits.

The sample mean and range are calculated and shown below:

Sample	X1	X2	X3	Sample mean(X1+X2+X3)/3	Sample Range
1	31	42	28	33.67 {(31+42+28)/3=33.67}	14
2	26	18	35	26.33	17
3	25	30	34	29.67	9
4	17	25	21	21.00	8
5	38	29	35	34.00	9
6	41	42	36	39.67	6
7	21	17	29	22.33	12
8	32	26	28	28.67	6
9	41	34	33	36.00	8
10	29	17	30	25.33	13
11	26	31	40	32.33	14
12	23	19	25	22.33	6
13	17	24	32	24.33	15
14	43	35	17	31.67	26
15	18	25	29	24.00	11
16	30	42	31	34.33	12
17	28	36	32	32.00	8
18	40	29	31	33.33	11
19	18	29	28	25.00	11
20	22	34	26	27.33	12
21	18	24	30	24.00	12
22	24	28	19	23.67	9
23	27	32	20	26.33	12
24	39	26	19	28.00	20
25	17	32	27	25.33	15
			Total	710.67	296
			Average	28.43	11.84

  $\overline{\overline{\text{x}}}=28.43$

  $\overline{\text{R}}=11.84$

  $n=3$

From the 3-sigma standard control chart, the coefficient values are obtained as follows:

  $\begin{array}{l}{A}_2=1.023\hfill \\ D_3=0\hfill \\ D_4=2.574\hfill \end{array}$

And the control limits are calculated

  $Uc{c}_{le}=\overline{\overline{\text{x}}}+A_2.\overline{\text{R}}=28.43+\left(1.023\times 11.84\right)=40.54$

  $Le{l}_a=\overline{\overline{\text{x}}}-A_2.\overline{\text{R}}=28.43-\left(1.023\times 11.84\right)=16.32$

  $UC{L}_R=D_4.\overline{\text{R}}=2.574\times 11.84=30.48$

  $LC{L}_R=D_3.\overline{\text{R}}=0\times 11.84=0$

The $\overline{\text{x}}$ and R charts are plotted with the above values:

The mean and range values are well within the control limits and hence the process is in control.