1. IENG 486: Statistical Quality & Process Control
12/4/2018
IENG Lecture 14
X-bar & s Charts:
Trial Limits & Standard Limits,
Control Chart Operating Characteristics
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

2. IENG 486: Statistical Quality & Process Control
Assignment
Reading: CH 8
8.1 – 8.3.2
8.3.4
8.7.1
Homework:
CH 6 Textbook Problems:
1; 6 a, b, d only – use Spreadsheet Template (Mat’ls pg)
11; 20 a only; 24; 30
12/4/2018
IENG 486: Statistical Quality & Process Control

3. R - (Range) Control Chart X-Bar (Means) Control Chart
X-Bar & R-Charts
The X-Bar Chart checks variability in location between samples
The R-Chart checks for changes in sample variation
R - (Range) Control Chart
LCL
UCL
Sample Number R
X-Bar (Means) Control Chart x
12/4/2018
IENG 486: Statistical Quality & Process Control

4. IENG 486: Statistical Quality & Process Control
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
12/4/2018
IENG 486: Statistical Quality & Process Control

5. IENG 486: Statistical Quality & Process Control
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
12/4/2018
IENG 486: Statistical Quality & Process Control

6. Operating Characteristic (OC) Curve
IENG 486: Statistical Quality & Process Control
12/4/2018
Operating Characteristic (OC) Curve
Ability of the x and R charts to detect shifts (sensitivity) is described by OC curves
For x chart; say we know s
Mean shifts from m (in-control value) to m1 = m0 +ks (out-of-control value)
The probability of NOT detecting the shift on the first sample after shift is
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

7. Ex. Probability of NOT Detecting Shift
IENG 486: Statistical Quality & Process Control
12/4/2018
Ex. Probability of NOT Detecting Shift
A 3-sigma x chart is used to monitor a normally distributed quality characteristic. The process std dev is 1.2 and the sample size is 5. The process mean is in-control at 22.
Find the probability that a shift to 24.4 is not detected on the first sample after the shift.
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

8. IENG 486: Statistical Quality & Process Control
12/4/2018
OC Curve for x Chart
Plot of b vs. shift size (in std dev units) for various sample sizes n
x chart not effective for small shift sizes, i.e., k  1.5s
Performance gets better for larger n and L or larger shifts
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

9. IENG 486: Statistical Quality & Process Control
12/4/2018
OC curve for R Chart
Uses distribution of relative range r.v., i.e.,
Suppose
s0 - in-control std dev
s1 - out-of-control std dev
OC curve for R chart plots b vs. ratio of in-control to out-of-control standard deviation for various sample sizes
That is, plot β vs. l = s1/s0
R chart not very effective for detecting shifts for small sample sizes
(see Fig in text)
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

10. Probability of Detecting Shift for Subsequent Samples
IENG 486: Statistical Quality & Process Control
12/4/2018
Probability of Detecting Shift for Subsequent Samples
After the shift has occurred:
P(NOT detecting shift ON 1st sample)
P(DETECTING shift ON 1st sample)
P(DETECTING shift ON 2nd sample)
P(DETECTING shift ON rth sample)
P(DETECTING shift BY 2nd sample)
P(DETECTING shift BY rth sample)
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

11. Average Run Length (ARL)
IENG 486: Statistical Quality & Process Control
12/4/2018
Average Run Length (ARL)
Expected number of samples taken before shift is detected is called the Average Run Length (ARL)
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

12. Performance of Any Shewhart Control Chart
IENG 486: Statistical Quality & Process Control
12/4/2018
Performance of Any Shewhart Control Chart
In-Control ARL:
Average number of points plotted on control chart before a false alarm occurs (ideally, should be large)
Out-of-Control ARL:
Average number of points, after the process goes out-of-control, before the control chart detects it (ideally, should be small)
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

13. IENG 486: Statistical Quality & Process Control
12/4/2018
ARL Curve for x Chart
Plot of ARL1 vs. shift size (in sd units) for various sample sizes n:
Average Time to Signal, (ATS):
Number of time periods that occur until signal is generated on control chart
h - time interval between samples
12/4/2018
IENG 486: Statistical Quality & Process Control
(c) , D.H. Jensen

14. IENG 486: Statistical Quality & Process Control
Next Assignment
Reading: CH 7, CH 6
8.1 – 8.3.2,
8.3.4,
8.7.1
Get Tables 8.2 & 8.3 for your engineering notebook
Homework:
CH 8 Textbook Problems:
9, 10, 25
12/4/2018
IENG 486: Statistical Quality & Process Control