Question

In a production process, eight samples of size 4 are collected and their means and ranges are given below. Construct mean chart and range chart with control limits.

Samples number	1	2	3	4	5	6	7	8
`bar"X"`	12	13	11	12	14	13	16	15
R	2	5	4	2	3	2	4	3

Answer 1

Samples number	1	2	3	4	5	6	7	8	Total
`bar"X"`	12	13	11	12	14	13	16	15	106
R	2	5	4	2	3	2	4	3	25

Let us A₂ = 0.73 and D₄ = 2.282 and D₃ = 0

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

`\overset{==}{"X"} = (sum"X")/"No. of samples" = 106/8` = 13.25

`bar"R" = (sum"R")/"No. of samples" = 25/8` = 3.125 = 3.12

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

= 13.25 + (0.73)(3.12)

= 13.25 + 2.2776 = 15.5276

= 15.53

CL = `\overset{==}{"X"}` = 13.25

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

= 13.25 – (0.73)(3.12)

= 13.25 – 2.2776 = 10.972

= 10.97

The control limits for Range chart is

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

= 2.28(3.12) = 7.11984

= 7.12

CL = `bar"R"` = 3.12

LCL = `"D"_3bar"R"` = 0(3.12) = 0