Question

From a process known to be in control, 6 samples of 4 units each were taken at random intervals and the units in the samples were weighed. The mean (Xbar) and range (R) for each of the six samples are given in the following table.

Sample	Mean	Range
1	5.2	0.7
2	4.6	1.1
3	4.1	1.2
4	4.7	1.1
5	4.6	0.8
6	4.4	0.9

a. Calculate the 3-sigma Xbar-chart and R-chart control limits.

b. Calculate the mean (Xbar) and range (R) for the following sample, which was taken from the same process at a later time.

Item number	1	2	3	4
Weight	4.7	5.0	5.1	4.4

Based on this sample and the control chart limits that you calculated in part (a), is the process in control? Why or why not?

Answer 1

Sample	Mean	Range
1	5.2	0.7
2	4.6	1.1
3	4.1	1.2
4	4.7	1.1
5	4.6	0.8
6	4.4	0.9

X double bar = sum of mean /N =27.6/6 =4.6

R bar= sum of range /N=5.8/6=0.97

Sample	Mean	Variance
1	5.2	0.36
2	4.6	0
3	4.1	0.25
4	4.7	0.01
5	4.6	0
6	4.4	0.04
	Sum	0.66

Standard deviation =Variance ^0.5 =0.66^0.5 =0.812

UCL for X bar = X double bar+3*Std. Deviation =4.6+3*0.812=7.036

LCL for X bar = X double bar-3*Std. Deviation =4.6-3*0.812=2.164

For R Chart,

Sample size of 6, D4=2, D3=0

LCL for R chart = D3*Rbar=0*0.97=0

UCL= D4*R bar=2*0.97=1.94

If we look at the mean values, the process seems to be in control as all the data falls within the UCL and LCL range, same is true for R chart

Answer b= X bar =(4.7+5+5.1+4.4)/4=4.8

Range=5.1-4.4=0.7

This process is also in control as all the data falls within the control limits