Question

Q6. A quality analyst took the following samples as a pilot study to construct the mean and range charts. The data below shows the weight in ounces. Construct the mean and range charts showing all control (action) –upper and lower limits, and warning upper and lower limits.

Sample Number	Item1	Item2	Item3	Item4
Sample 1	11.9	12.5	12.4	12.7
Sample 2	12.5	12.5	12.4	12.8
Sample 3	12.2	12.8	12.7	12
Sample 4	12.3	12.4	12.8	12.4
Sample 5	12.1	12.8	12.4	11.9
Sample 6	12.6	12.4	12.1	12.3

The following day, the analyst took another 6 samples to check if the process under control or not. Theses samples are as follows:

Sample Number	Item1	Item2	Item3	Item4
Sample 1	11.9	11.5	12.4	12.3
Sample 2	12.5	11.5	13.4	11.8
Sample 3	12.2	11.8	11.7	12
Sample 4	14.3	11.4	11.8	13.4
Sample 5	15.1	11.8	12.4	10.9
Sample 6	11.6	10.4	10.1	15.3

Is the process under control? Why? What actions should you take?

Factors for calculating control limits for control charts

For MEANS

Sample size (n)	Constant (dn)	Factors for warning limits 1.96/dn√n	Factors for action limits 3.09√n
2	1.128	1.23	1.94
3	1.693	0.67	1.05
4	2.059	0.48	0.75
5	2.326	0.38	0.59
6	2.334	0.32	0.50
7	2.704	0.27	0.43
8	2.847	0.24	0.38
9	2.970	0.22	0.35
10	3.078	0.20	0.32

Factors for calculating control limits for control charts

For RANGES

Sample size	Upper warning limit	Lower warning limit	Upper action limit	Lower action limit
2	2.81	0.04	4.12	0.00
3	2.17	0.18	2.98	0.04
4	1.93	0.29	2.57	0.10
5	1.81	0.37	2.34	0.16
6	1.72	0.42	2.21	0.21
7	1.66	0.46	2.11	0.26
8	1.62	0.50	2.04	0.29
9	1.58	0.52	1.99	0.32
10	1.56	0.54	1.94	0.35

Answer 1

Sample No.	Item1	Item2	Item3	Item4	Sample Mean	Sample Range
Sample 1	11.90	12.50	12.40	12.70	12.38	0.80
Sample 2	12.50	12.50	12.40	12.80	12.55	0.40
Sample 3	12.20	12.80	12.70	12.00	12.43	0.80
Sample 4	12.30	12.40	12.80	12.40	12.48	0.50
Sample 5	12.10	12.80	12.40	11.90	12.30	0.90
Sample 6	12.60	12.40	12.10	12.30	12.35	0.50
				Avg.	12.4125	0.6500
	Mean chart	Range chart
Sample size	4	4
Action limits factor	0.75
UCL	12.900	1.671
LCL	11.925	0.065
CL	12.413	0.650
Warning limits factor	0.48
UAL	12.725	1.255
LAL	12.101	0.189

Using the above [LCL, UCL] and [LAL, UAL] ranges, plot the second set of samples as follows:

Sample Number	Item1	Item2	Item3	Item4	Sample Mean	Sample Range
Sample 1	11.90	11.50	12.40	12.30	12.03	0.90
Sample 2	12.50	11.50	13.40	11.80	12.30	1.90
Sample 3	12.20	11.80	11.70	12.00	11.93	0.50
Sample 4	14.30	11.40	11.80	13.40	12.73	2.90
Sample 5	15.10	11.80	12.40	10.90	12.55	4.20
Sample 6	11.60	10.40	10.10	15.30	11.85	5.20

The process is out of control both in mean and range charts.