1. Control Charts

2. Statistical Process Control
The objective of a process control system is to provide a statistical signal when assignable causes of variation are present

3. Control Charts
Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes
Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them.

4. Steps In Creating Control Charts
Take samples from the population and compute the appropriate sample statistic
Use the sample statistic to calculate control limits and draw the control chart
Plot sample results on the control chart and determine the state of the process (in or out of control)
Investigate possible assignable causes and take any indicated actions
Continue sampling from the process and reset the control limits when necessary

5. Control Charts for Variables
For variables that have continuous dimensions
Weight, speed, length, strength, etc.
x-charts are to control the central tendency of the process
R-charts are to control the dispersion of the process
These two charts must be used together

6. x-charts For x-Charts when we know s
Upper control limit (UCL) = x + zsx
Lower control limit (LCL) = x - zsx
where x = mean of the sample means or a target value set for the process
z = number of normal standard deviations
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size

7. Sample average of 9 boxes
x-charts - Example
The weights of boxes of Oat Flakes within a large production lot are sampled each hour. 12 different samples where selected and weighted and the average of each sample is presented in the following table. Each sample contains 9 boxes and the standard deviation of the population is 1. Managers want to set control limits that include 99.73% of the sample mean.
Hour
Sample average of 9 boxes 1
16.1 7
15.2 2
16.8 8
16.4 3
15.5 9
16.3 4
16.5 10
14.8 5 11
14.2 6 12
17.3

8. x-charts - Example For 99.73% control limits, z = 3
UCLx = x + zsx = (1/3) = 17 ounces
LCLx = x - zsx = (1/3) = 15 ounces

9. Variation due to natural causes
x-charts - Example
Control Chart for sample of 9 boxes
Variation due to assignable causes
Out of control
Sample number
| | | | | | | | | | | |
17 = UCL
15 = LCL
16 = Mean
Variation due to natural causes
Out of control

10. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Normal behavior. Process is “in control.”

11. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
One plot out above (or below). Investigate for cause. Process is “out of control.”

12. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Trends in either direction, 5 plots. Investigate for cause of progressive change.

13. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Two plots very near lower (or upper) control. Investigate for cause.

14. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Run of 5 above (or below) central line. Investigate for cause.

15. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
Erratic behavior. Investigate.

16. x-charts For x-Charts when we don’t know s
Upper control limit (UCL) = x + A2R
Lower control limit (LCL) = x - A2R
where R = average range of the samples
A2 = control chart factor found in the Table in the next slide
x = mean of the sample means

17. Control Chart Factors (3 sigma)
Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3

18. x-charts - Example
Super Cola bottles soft drinks labeled “net weight 12 ounces”. Indeed, an overall process average of 12 ounces has been found by taking many samples, in which each sample contained 5 bottles. The average range of the process is 0.25 ounces. We want to determine the upper and lower control limits for averages in this process.

19. x-charts - Example Process average x = 12 ounces Average range R = .25
Sample size n = 5
UCL =
Mean = 12
LCL =
UCLx = x + A2R
= 12 + (.577)(.25) =
= ounces
LCLx = x - A2R =
= ounces

20. R–Chart Type of variables control chart Shows sample ranges over time
Difference between smallest and largest values in sample
Monitors process variability
Independent from process mean

21. R–Chart For R-Charts Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors from the previous Table - Control Chart Factors (3 sigma)

22. R–Chart - Example
The average range of a product at the National Manufacturing Co. is 5.3 pounds. With a sample size of 5, the owners want to determine the upper and lower control chart limits for the range
UCL = 11.2
Mean = 5.3
LCL = 0
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
LCLR = D3R
= (0)(5.3)
= 0 pounds

23. Mean and Range Charts (a)
These sampling distributions result in the charts below
(Sampling mean is shifting upward but range is consistent)
x-chart
(x-chart detects shift in central tendency)
UCL
LCL
R-chart
(R-chart does not detect change in mean)
UCL
LCL

24. Mean and Range Charts (b)
These sampling distributions result in the charts below
(Sampling mean is constant but dispersion is increasing)
x-chart
(x-chart does not detect the increase in dispersion)
UCL
LCL
R-chart
(R-chart detects increase in dispersion)
UCL
LCL