1. IENG 486: Statistical Quality & Process Control
11/14/2018
IENG Lecture 13
Statistical Basis for
X-Bar & R Control Charts
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

2. IENG 486: Statistical Quality & Process Control
Teminology
Causes of Variation:
Assignable Causes
Keep the process from operating predictably
Things that we can do something about
Common Causes
Random, inherent variation in the process
Meaning of Control:
In Specification
Meets customer constraints on product
In Statistical Control
No Assignable Causes of variation present in the process
11/14/2018
IENG 486: Statistical Quality & Process Control

3. IENG 486: Statistical Quality & Process Control
11/14/2018
Assignment:
Reading:
CH5: 5.3 (already read & 5.4)
Start on CH6: all except & 6.4
Homework 4:
Textbook Problems CH5: 9, 11, 13, 23, & 24
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

4. Charts Based on Standard Values, x Chart
IENG 486: Statistical Quality & Process Control
11/14/2018
Charts Based on Standard Values, x Chart
If values for m and s are known (i.e., do not need to estimate from data)
Quantity A is tabulated in Appendix VI
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

5. R - Chart Based on Standard Values
IENG 486: Statistical Quality & Process Control
11/14/2018
R - Chart Based on Standard Values
If values for R and s are known
r.v. W = R / s – relative range
The parameters of the distribution of W are a function of n
From the relative range we can compute the mean of R
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

6. The Standard Deviation of R
IENG 486: Statistical Quality & Process Control
11/14/2018
The Standard Deviation of R
The standard deviation of R is given as: (Text does not derive this)
therefore
where
D1 and D2 are constants tabulated in Appendix VI
Caution: Be careful when using standard values
make sure these values are representative of the actual process
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

7. Computing Trial Control Limits (from sample data) for x Chart
IENG 486: Statistical Quality & Process Control
11/14/2018
Computing Trial Control Limits (from sample data) for x Chart
General model for x chart
Substituting estimates for μx and σx and using 3-sigma limits:
Where A2 comes from Appendix VI and depends on n
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

8. Computing Trial Control Limits for R - Chart
IENG 486: Statistical Quality & Process Control
11/14/2018
Computing Trial Control Limits for R - Chart
x and R charts come as a pair
General model for R chart
Substituting estimates for mR and sR and using 3-sigma limits
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

9. Computing Trial Control Limits for R - Chart (continued)
IENG 486: Statistical Quality & Process Control
11/14/2018
Computing Trial Control Limits for R - Chart (continued)
where and
D3 and D4 are tabulated in Appendix VI and
depend on n
NOTE: R chart is quite sensitive to departures from normality
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

10. x & R Trial Control Chart Limits: Guidelines for Sampling
IENG 486: Statistical Quality & Process Control
11/14/2018
x & R Trial Control Chart Limits: Guidelines for Sampling
Sample should be of size 3 to 8 (sizes 4 – 6 are more common)
Sample must be homogeneous
same time (consecutive units)
same raw materials
same operator
same machine
Time may pass between samples but not within samples
11/14/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen & R. C. Wurl

11. Steps for Trial Control Limits
Start with 20 to 25 samples
Use all data to calculate initial control limits
Plot each sample in time-order on chart.
Check for out of control sample points
If one (or more) found, then:
Investigate the process;
Remove the special cause; and
Remove the special cause point and recalculate control limits.
If can’t find special cause - drop point & recalculate anyway
11/14/2018
IENG 486: Statistical Quality & Process Control

12. Control Chart Sensitizing Rules
Western Electric Rules:
One point plots outside the three-sigma limits;
Two out of three consecutive points plot beyond two-sigma warning limits on the same side of the center line;
Four out of five consecutive points plot beyond one-sigma warning limits on the same side of the center line; or
Eight consecutive points plot on one side of the center line.
If chart shows lack of control, investigate for special cause
11/14/2018
IENG 486: Statistical Quality & Process Control

13. Control Chart Examples
LCL
UCL x
LCL
UCL x
LCL
UCL x
LCL
UCL x
Rule 1
Rule 2
Rule 3
Rule 4
11/14/2018
IENG 486: Statistical Quality & Process Control

14. Control Chart Sensitizing Rules
Additional Sensitizing Rules:
One or more points very near a control limit.
Six points in a row steadily increasing or decreasing.
Eight points in a row on both sides of the center line, but none in-between the one-sigma warning limits on both sides of the center line.
Fourteen points in a row alternating above and below the center line.
Fifteen points in a row anywhere between the one-sigma warning limits (including either side of the center line).
Any unusual or non-random pattern to the plotted points.
11/14/2018
IENG 486: Statistical Quality & Process Control