Where do Control Chart Coefficients come from?
Process Quality Characteristic: Normal Distribution
Process Mean = 200, Process Sigma = 10
10
170
180
190
200
210
220
230
X measurement
Subgroups of a given size are selected from this hypothetical process distribution. For
each subgroup a subgroup range, R, is calculated. Every subgroup will have different
values and thus produce different ranges. If we think about taking many, many
subgroups over time we will build up a distribution of values of R. This is called the
sampling distribution of R. In order to see where the control chart coefficients come
from, we need to look at the sampling distribution of R for each subgroup size.
The center of the sampling distribution of R is related to the process standard deviation;
Mean R = d2*s
Most of the sampling distribution of R will fall between;
D3*Mean R and D4*Mean R
Simulated sampling distribution of R, n=2
0.06
0
Mean R
D_4*Mean R
d_2 = 1.128
0.05
Density
0.04
0.03
0.02
0.01
0.00
0
10
20
30
40
50
60
70
80
Subgroup Range, R
Simulated sampling distribution of R, n=4
D_4*Mean R
Mean R
0
0.06
d_2 = 2.059
0.05
Density
0.04
0.03
0.02
0.01
0.00
0
10
20
30
40
50
Subgroup Range, R
60
70
80
Simulated sampling distribution of R, n=10
D_3*Mean R
Mean R
D_4*Mean R
0.06
d_2 = 3.078
0.05
Density
0.04
0.03
0.02
0.01
0.00
0
10
20
30
40
50
60
70
80
Subgroup Range, R
Simulated sampling distribution of X-bar, n=2
LCL
Center
UCL
0.15
Density
0.10
0.05
0.00
180
190
200
210
Subgroup Mean, X-bar
220
Simulated sampling distribution of X-bar, n=4
LCL
0.15
Center
UCL
Density
0.10
0.05
0.00
180
190
200
210
220
Subgroup Mean, X-bar
Simulated sampling distribution of X-bar, n=10
LCL
Center
190
200
UCL
0.15
Density
0.10
0.05
0.00
180
210
Subgroup Mean, X-bar
220
Cylinder Bore
s chart
1
10
Sample StDev
1
3.0SL=6.493
5
S=3.108
0
-3.0SL=0.00E+00
0
5
10
15
20
25
30
35
Sample Number
Sample Mean
Cylinder Bore
X-bar chart
205
3.0SL=204.7
200
X=200.2
-3.0SL=195.8
195
0
5
10
15
20
25
Sample Number
30
35