Question

Ten samples each of size five are drawn at regular intervals from a manufacturing process. The sample means `(bar"X")` and their ranges (R) are given below:

Sample number	1	2	3	4	5	6	7	8	9	10
`bar"X"`	49	45	48	53	39	47	46	39	51	45
R	7	5	7	9	5	8	8	6	7	6

Calculate the control limits in respect of `bar"X"` chart. (Given A₂ = 0.58, D₃ = 0 and D₄ = 2.115) Comment on the state of control

Answer 1

Sample number	1	2	3	4	5	6	7	8	9	10	Total
`bar"X"`	49	45	48	53	39	47	46	39	51	45	462
R	7	5	7	9	5	8	8	6	7	6	68

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

`\overset{==}{"X"} = (sumbar"X")/"No. of samples" = 462/10` = 46.2

`bar"R" = (sum"R")/"No. of samples" = 68/10` = 6.8

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

= 46.2 + (0.58)(6.8)

= 46.2 + 3.944

= 50.144

= 50.14

CL = `\overset{==}{"X"}` = 46.2

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

= 46.2 – (0.58)(6.8)

= 46.2 – 3.944

= 42.256

= 42.26

The control limits for Range chart is

UCL = `"D"_4 bar"R"`

= (2.115)(6.8)

= 14.382

= 14.38

CL = `bar"R"` = 6.8

LCL = `"D"_3bar"R"` = (0)(6.8) = 0

Since all individual sample range points fall within the control limits, the process variability is in a state of statistical control.