Question

A machine is set to deliver packets of a given weight. Ten samples of size five each were recorded. Below are given relevant data:

Sample number	1	2	3	4	5	6	7	8	9	10
`bar"X"`	15	17	15	18	17	14	18	15	1	16
R	7	7	4	9	8	7	12	4	11	5

Calculate the control limits for mean chart and the range chart and then comment on the state of control, (conversion factors for n = 5, A₂ = 0.58, D₃ = 0 and D₄ = 2.115)

Answer 1

Sample number	1	2	3	4	5	6	7	8	9	10	Tota
Mean `(bar"X")`	15	17	15	18	17	14	18	15	1	16	`sum"X"` = 162
Range (R)	7	7	4	9	8	7	12	4	11	5	`sum"R"` = 74

The control limits for `bar"X"` chart is 

`\overset{==}{"X"} = (sum"X")/"No. of samples" = 162/10` = 16.2

`bar"R" = (sum"R")/"No. of samples" = 74/10` = 7.4

UCL = `\overset{==}{"X"} + "A"_2+ bar"R"`

= 16.2 + (0.58)(7.4)

= 16.2 + 4.292

= 20.492

= 20.49

CL = `\overset{=}{"X"}` = 16.2

LCL = `\overset{==}{"X"} - "A"_2 bar"R"`

= 16.2 – (0.58)(7.4)

= 16.2 – 4.292

= 11.908

= 11.91

The control limits for Range chart is

CL = `"D"_4bar"R"`

= (2.115)(7.4)

= 15.651

= 15.65

CL = `bar"R"` = 7.4

LCL = `"D"_3 bar"R"` = (0)(7.4) = 0

The above diagram shows all the three control lines with the data points plotted.

We see that all the points of the sample mean are within the control limits.

We now draw the R chart for the given data.

The above diagram shows all the three control lines with the sample range points plotted.

We observe that all the points are within the control limits.

Conclusion: From the above two plots of the sample mean `bar"X"` and sample range R, we conclude that the process is in control.