Question

The following data show the values of sample mean `(bar"X")` and its range (R) for the samples of size five each. Calculate the values for control limits for mean, range chart and determine whether the process is in control.

Sample Number	1	2	3	4	5	6	7	8	9	10
Mean	11.2	11.8	10.8	11.6	11.0	9.6	10.4	9.6	10.6	10.0
Range	7	4	8	5	7	4	8	4	7	9

(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	Total
Mean `bar"X"`	11.2	11.8	10.8	11.6	11.0	9.6	10.4	9.6	10.6	10.0	106.6
Range (R)	7	4	8	5	7	4	8	4	7	9	63

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

`\overset{==}{"X"} = (sumbar"X")/"Number of samoles" = 106.6/10` = 10.66

`bar"R" = (sum"R")/"R" = 63/10` = 6.3

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

= 10.66 + (0.58)(6.3)

= 10.66 + 3.654 = 14.314

= 14.31

CL = `\overset{==}{"X"}` = 10.66

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

= 10.66 – (0.58)(6.3)

= 10.66 – 3.654

= 7.006

The control limits for Range chart is

UCL = `"D"_4 bar"R"` = 2.115(6.3)`

= 13.3245

= 13.32

CL = `bar"R"` = 6.3

LCL = `"D"_3 bar"R"` = 0(6.3) = 0

Conclusion: Since all the points of sample range is within UCL of R chart, the process is in control.