Question

The thickness of a metal part is an important quality parameter. Data on thickness (in inches) are given in the following table, for 25 samples of five parts each.

Sample Number	x1	x2	x3	x4	x5
1	0.0629	0.0636	0.0640	0.0634	0.0641
2	0.0630	0.0632	0.0620	0.0624	0.0627
3	0.0628	0.0631	0.0633	0.0633	0.0630
4	0.0634	0.0630	0.0631	0.0632	0.0633
5	0.0619	0.0628	0.0630	0.0619	0.0625
6	0.0613	0.0629	0.0634	0.0625	0.0628
7	0.0630	0.0639	0.0625	0.0629	0.0627
8	0.0628	0.0627	0.0622	0.0625	0.0627
9	0.0623	0.0626	0.0633	0.0630	0.0624
10	0.0631	0.0631	0.0633	0.0631	0.0630
11	0.0635	0.0630	0.0638	0.0635	0.0633
12	0.0623	0.0630	0.0630	0.0627	0.0629
13	0.0635	0.0631	0.0630	0.0630	0.0630
14	0.0645	0.0640	0.0631	0.0640	0.0642
15	0.0619	0.0644	0.0632	0.0622	0.0635
16	0.0631	0.0627	0.0630	0.0628	0.0629
17	0.0616	0.0623	0.0631	0.0620	0.0625
18	0.0630	0.0630	0.0626	0.0629	0.0628
19	0.0636	0.0631	0.0629	0.0635	0.0634
20	0.0640	0.0635	0.0629	0.0635	0.0634
21	0.0628	0.0625	0.0616	0.0620	0.0623
22	0.0615	0.0625	0.0619	0.0619	0.0622
23	0.0630	0.0632	0.0630	0.0631	0.0630
24	0.0635	0.0629	0.0635	0.0631	0.0633
25	0.0623	0.0629	0.0630	0.0626	0.0628

(a) Using all the data find trial control limits for X¯ and R charts. Round your answers to 5 decimal places (e.g. 98.76543).

X¯ Control Limits:

UCL =

CL =

LCL =

R Control Limits:

UCL =

CL =

LCL =

(b) Construct X¯ and R control charts using the control limits from part (a) to identify out-of-control points. (Use Minitab, Excel or any other statistical software.)

Does the process appear to be in control or out-of-control?

How many of the subgroups fall outside the control limits on both charts combined. (If none are outside the limits, type "0".)

Answer 1

a) -

To calculate limits for & R charts, we have to calculate mean & range of each sample -

( In excel, you can use AVERAGE function to calculate mean)

R = Maximium observation - Minimum observation (In excel, you can use - MAX() - MIN() )

Observation table -

Sample	x1	x2	x3	x4	x5	x_bar	Range
1	0.0629	0.0636	0.064	0.0634	0.0641	0.0636	0.0012
2	0.063	0.0632	0.062	0.0624	0.0627	0.06266	0.0012
3	0.0628	0.0631	0.0633	0.0633	0.063	0.0631	0.0005
4	0.0634	0.063	0.0631	0.0632	0.0633	0.0632	0.0004
5	0.0619	0.0628	0.063	0.0619	0.0625	0.06242	0.0011
6	0.0613	0.0629	0.0634	0.0625	0.0628	0.06258	0.0021
7	0.063	0.0639	0.0625	0.0629	0.0627	0.063	0.0014
8	0.0628	0.0627	0.0622	0.0625	0.0627	0.06258	0.0006
9	0.0623	0.0626	0.0633	0.063	0.0624	0.06272	0.001
10	0.0631	0.0631	0.0633	0.0631	0.063	0.06312	0.0003
11	0.0635	0.063	0.0638	0.0635	0.0633	0.06342	0.0008
12	0.0623	0.063	0.063	0.0627	0.0629	0.06278	0.0007
13	0.0635	0.0631	0.063	0.063	0.063	0.06312	0.0005
14	0.0645	0.064	0.0631	0.064	0.0642	0.06396	0.0014
15	0.0619	0.0644	0.0632	0.0622	0.0635	0.06304	0.0025
16	0.0631	0.0627	0.063	0.0628	0.0629	0.0629	0.0004
17	0.0616	0.0623	0.0631	0.062	0.0625	0.0623	0.0015
18	0.063	0.063	0.0626	0.0629	0.0628	0.06286	0.0004
19	0.0636	0.0631	0.0629	0.0635	0.0634	0.0633	0.0007
20	0.064	0.0635	0.0629	0.0635	0.0634	0.06346	0.0011
21	0.0628	0.0625	0.0616	0.062	0.0623	0.06224	0.0012
22	0.0615	0.0625	0.0619	0.0619	0.0622	0.062	0.001
23	0.063	0.0632	0.063	0.0631	0.063	0.06306	0.0002
24	0.0635	0.0629	0.0635	0.0631	0.0633	0.06326	0.0006
25	0.0623	0.0629	0.063	0.0626	0.0628	0.06272	0.0007
						=0.06294	=0.00094

So, we get - To find & , we can take average of all & R values.

=0.06294 & =0.00094

Control limits for chart are -

LCL = - A₃ = 0.06294 - (1.427*0.00094) = 0.06294 - 0.00134 = 0.0616

CL = = 0.06294

UCL = + A₃ = 0.06294 + (1.427*0.00094) = 0.06294 + 0.00134 = 0.06428

Control limits for R-chart are -

LCL = D₃ = 0*0.00094 = 0

CL = = 0.00094

UCL = D₄ = 2.114*0.00094 = 0.00199

Values of A₃, D₃ & D₄ are taken from table given below. Values taken corresponding to 5 for all constants, since our sample size is 5 for each sample.

b) -

Control chart -

In control chart, none of the value is outside the control limits. So, we can say that process is under control.

R control chart-

In this chart, we can see that two points are above the upper control limits. Sample no. 6 & sample no. 15 fall outside the control limits. So, the process is out of control.