1. Special Control Charts II
Review
X-bar, R, S

2. X-Bar & Sigma-Charts Used when sample size is greater than 10
X-Bar Control Limits:
Approximate 3 limits are found from S & table
Sigma-Chart Control Limits:
Approximate, asymmetric 3 limits from S & table

3. X-Bar & Sigma-Charts
Limits can also be generated from historical data:
X-Bar Control Limits:
Approximate 3 limits are found from known 0 & table
Sigma-Chart Control Limits:
Approximate, asymmetric 3 limits from 0 & table

4. X-bar and S charts
Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R
S chart limits:
UCL = B6σ = B4*S-bar
Center Line = c4σ = S-bar
LCL = B5σ = B3*S-bar
X-bar chart limits
UCL = X-doublebar +A3S-bar
Center line = X-doublebar
LCL = X-doublebar -A3S-bar

6. Special Variables Control Charts
x-chart for individuals
Moving ranged control chart
QSUM chart

7. Individual Measurements
IE 355: Quality and Applied Statistics I
Individual Measurements
Sometimes repeated measures in a subsample don't make sense:
inventory level
accounts payable – price of an item
Other reasons for using individual measurements
Variation in sample only reflects measurement error, e.g., batch production of chemicals
Automated inspection – every unit is analyzed
Production rate very slow – inconvenient to wait for large enough sample
(c) D.H. Jensen

8. Control Charts for Individual Measurements
IE 355: Quality and Applied Statistics I
Control Charts for Individual Measurements
Notes about Individuals Charts:
Sample points must be relatively frequent
There is more sampling error (false alarm & insensitivity)
Sample points tend to be non-normal
points are not averages and central limit theorem does not apply
Control Chart Types for Individuals:
Shewhart x-chart and Moving Range chart
MA – Moving Average chart
EWMA – Exponentially Weighted Moving Average
CUSUM – Cumulative Sum
(c) D.H. Jensen

9. Moving Range Control Chart Viscosity of Paint Primer
IE 355: Quality and Applied Statistics I
Moving Range Control Chart Viscosity of Paint Primer
MR2i = |xi – xi-1| or MR3i = |xi – xi-2|
Computation of the Moving Range:
Obs i xi
MR2
MR3 1
33.75 - 2
33.05
0.70 3
34.00
0.95
0.25 4
33.81
0.19
0.76 5
33.46
0.35
0.46 … n
Average
33.52
0.48
0.42
(c) D.H. Jensen

10. Moving Range Control Chart, Cont'd
IE 355: Quality and Applied Statistics I
Moving Range Control Chart, Cont'd
General model for moving range chart:
Plot MR2i = |xi – xi-1| or MR3i = |xi – xi-2| on control chart
Substituting estimates for mR and sR and using “3-sigma” limits:
Where MR is:
(c) D.H. Jensen

11. Moving Range Control Chart, Cont'd
IE 355: Quality and Applied Statistics I
Moving Range Control Chart, Cont'd
For MR2 use d2 for n = 2 For MR3 use d2 for n = 3
Very similar to Range chart except we’re using moving range instead of average range
(c) D.H. Jensen

12. x Control Chart (Individual Measurements Chart)
IE 355: Quality and Applied Statistics I
x Control Chart (Individual Measurements Chart)
Plot sample statistic: x
General model for x chart
Substituting estimates for mx and sx and using 3-sigma limits
where:
(c) D.H. Jensen

13. IE 355: Quality and Applied Statistics I
x Control Chart Cont'd
x chart upper and lower limits:
(c) D.H. Jensen

14. Cautions for x & Moving Range charts:
IE 355: Quality and Applied Statistics I
Cautions for x & Moving Range charts:
Always check x’s for normality
If x’s not normal, control limits are inappropriate
Use zone rules ONLY if the x’s are Normal
Very BAD at detecting small shifts, i.e., shifts < 2s
x chart (n = 5)
x chart (n = 1)
size of shift b
ARL1 1s
0.78
5.25
0.98
43.96 2s
0.07
1.08
0.84
6.30 3s
0.00
1.00
0.50
2.00
(c) D.H. Jensen

15. IE 355: Quality and Applied Statistics I
Montgomery(5th ed.) Example 5-5, p. 250 Viscosity of Aircraft Primer Paint
Plot MR2i = |xi – xi-1| on control chart
From initial data compute x and MR:
(since d2 = for n =2)
Control Limits for Moving Range chart (use D4 & D3 for n =2)
(c) D.H. Jensen

16. Ex.: Viscosity of Aircraft Primer Paint, Cont'd
IE 355: Quality and Applied Statistics I
Ex.: Viscosity of Aircraft Primer Paint, Cont'd
Compute control Limits for x Chart
(c) D.H. Jensen

17. IE 355: Quality and Applied Statistics I
CUSUM Control Chart
Incorporates all the information in the sequence of sample values by
plotting the cumulative sums of the deviations of the sample values from a target value, m0
CUSUM can be used to monitor
process mean
defectives
defects
variance
CUSUM can have sample size n  1
We concentrate on sample size n = 1
(c) D.H. Jensen

18. Basic Principle of CUSUM
IE 355: Quality and Applied Statistics I
Basic Principle of CUSUM
Plot Ci – CUSUM sample statistic
Example: Say target m0 = 10
If the process remains in-control, Ci remains near 0
Obs i xi
(xi – m0) 1
9.45
-0.55 2
7.99
-2.01
-0.55 – 2.01 = -2.56 3
9.29
-0.71
-2.56 – 0.71 = -3.27 4
11.66
1.66
= -1.61 5
12.16
2.16
= 0.55
(c) D.H. Jensen

19. Tabular CUSUM Control Chart
IE 355: Quality and Applied Statistics I
Tabular CUSUM Control Chart
xi ~ N(m0, s) - quality characteristic
CUSUM works by compiling the statistics:
Ci+ = accumulated deviations above m0 (resets to 0 if it would go negative)
Ci– = accumulated deviations below m0 (resets to 0 if it would go positive)
The Tabular CUSUM
Record following values in table:
where starting values are
(c) D.H. Jensen

20. Tabular CUSUM Control Chart Cont'd
IE 355: Quality and Applied Statistics I
Tabular CUSUM Control Chart Cont'd
Let m1 = out-of-control value then
K is reference value chosen halfway between target m0 and out-of-control value
With shift expressed in std dev units, i.e.,
and accumulate deviations from μ0 that are greater than K
and are reset to zero upon becoming negative
(c) D.H. Jensen

21. How to Determine if Process Out-of-Control?
IE 355: Quality and Applied Statistics I
How to Determine if Process Out-of-Control?
H - decision interval
If or exceed the decision interval (H), the process is considered out-of-control
Rule of thumb value for H
Choose H to be five times the process standard deviation, H = 5s
Counters N+ and N– record the number of consecutive periods the CUSUM and rose above zero, respectively.
The counters can be used to indicate when the shift most likely occurred
(c) D.H. Jensen

22. IE 355: Quality and Applied Statistics I
Example
m0 =10, n =1, s = 1.0
Say magnitude of shift we want to detect is ds = 1 (1.0) = 1.0
Parameters:
m1 = m0 + ds = 10 + (1)(1) = 11
K = d/2 = ½ = 0.5
H = 5s = (5)(1) = 5
Counters N+ and N-
N+ records the number of consecutive periods since the upper-side cusum Ci+ rose above the value of zero
N- records the number of consecutive periods since the lower-side cusum Ci- rose above the value of zero
(c) D.H. Jensen

23. IE 355: Quality and Applied Statistics I
The upper-side cusum at period 29 is C29+ = 5.28
Since this is the first period at which Ci+ > H =5, we would conclude that the process is out of control at that point
Since N+ = 7 at period 29, we would conclude that the process was last in control at period 29-7=22, so the shift likely occurred between periods 22 and 23
(c) D.H. Jensen

24. Tabular CUSUM Example

25. Estimate of New Shifted Process Mean
IE 355: Quality and Applied Statistics I
Estimate of New Shifted Process Mean
Use this estimate to bring process back to the target value m0
e.g.: At period 29,
New process average estimate is
New process average estimate is:
Quality characteristic has shifted from 10 to
Make process adjustment to decrease value of quality characteristic by 1.25.
(c) D.H. Jensen

26. Notes about CUSUM control charts
IE 355: Quality and Applied Statistics I
Notes about CUSUM control charts
Do not apply zone rules
Do not apply run rules
Successive values of and are not independent
(c) D.H. Jensen

27. Recommendations for CUSUM design
IE 355: Quality and Applied Statistics I
Recommendations for CUSUM design
Let H = hs and K = ks where s is process std dev
Using h = 4 or h =5 and k = 1/2 gives CUSUM w/ good ARL
Shift (multiple of s)
ARL for h = 4
ARL for h = 5
168
465
0.25
74.2
139
0.50
26.6
38.0
0.75
13.3
17.0
1.00
8.38
10.4
1.50
4.75
5.75
2.00
3.34
4.01
2.50
2.62
3.11
3.00
2.19
2.57
4.00
1.71
2.01
(c) D.H. Jensen

28. CUSUM Chart
X-bar, R charts work well to detect significant shifts in the mean (1.5s - 2.0s)
However, suppose we wish to detect smaller shifts in the process
Consider a piston ring process with target value at 10.0 cm.
First 10 sampled at normal with m = 10
Second 10 sampled at normal with m = 11
Control limits computed at LCL = 7, UCL = 13

29. Example

30. Example

31. CUSUM Idea å å Show the cumulative effects of relatively small changes= ( x - m ) i j o j = 1 i - 1 å = ( x - m ) + ( x - m ) i o j o j = 1 = ( x - m ) + C i o i - 1

32. Example, CUSUM Table

33. Example; CUSUM Chart