1. 1 SMU EMIS 7364 NTU TO-570-N More Control Charts Material Updated: 3/24/04 Statistical Quality Control Dr. Jerrell T. Stracener, SAE Fellow

2. 2 Operating Characteristic (OC) Function for the x - Chart The OC curve describes the ability of the x-chart to detect shifts in process quality. For an x-chart with  known & constant mean  shifts from in-control value,  0 to another value  1, where  1 =  0 + K

3. 3 Operating Characteristic (OC) Function for the x – Chart (continued)

4. 4 where and L is usually 3, the three-sigma limits

5. 5 Example If n=5 & L=3, determine & plot the OC function vs K, where  1 =  0 + K.

6. 6 Example - Solution

7. 7 OC Function of the Fraction Nonconforming Control Chart

8. 8 Where [nUCL] denotes the largest integer  nUCL and denotes the smallest integer  nLCL Note: The OC curve provides a measure of the sensitivity of the control chart – i.e., its ability to detect a shift in the process fraction nonconforming from the nominal value p to some other value p.

9. 9 Example For a fraction nonconforming control chart with parameters n = 50, LCL = 0.0303, and UCL = 0.3697, Determine & plot the OC curve.

10. 10 Example - Solution

11. 11 Example - Solution p OC(p)

12. 12 OC Function for c-charts and u-charts For the c-chart

13. 13 OC Function for c-charts and u-charts For the u-chart

14. 14 Example Determine & plot the OC function for a u-chart with parameter. LCL = 6.48, and UCL = 32.22.

15. 15 Example - Solution

16. 16 Example - Solution u OC(u)

17. 17 Average Run Length for x-Charts Performance of Control Charts can be characterized by their run length distribution. Run Length (RL) of a control chart is defined to be the number of samples until the process characteristic exceeds the control limits for the first time. Run Length, RL, is a random variable and therefore has a probability distribution

18. 18 Average Run Length for x-Charts Let p = P(x falls outside control limits) Then P(RL = 1)= P( x 1 falls outside CL )=p P(RL = 2)= P( x 1 falls inside CL & x 2 falls outside of CL ) = (1-p)p P(RL = 3)= P( x 1, x 2 fall inside CL & x 3 falls outside of CL ) = (1-p)(1-p)p P(RL = i)= P( x 1, x 2, … x i-1 fall inside CL & x i falls outside of CL ) = (1-p) i-1 p

19. 19 Average Run Length for x-Charts Therefore, the probability mass function of RL is The mean or expected value of RL is

20. 20 Average Run Length for x-Charts The Average Run Length, ARL, indicates the number of samples needed, on the average before x will exceed the control chart limits.

21. 21 Probability of Out-Of-Control Signal and ARL Process in control with mean  0 p = 1 – P(LCL  x  UCL) = 0.0027 ARL i.e., one the average we would expect 1 out- of-control signal out of 370 samples.

22. 22 Probability of Out-Of-Control Signal and ARL Process in control with mean  1  0 +with constant  What happens if the process goes out of control? How long does it take until the control charts detects the shift? Probability of detecting shift

23. 23 Probability of Out-Of-Control Signal and ARL

24. 24 Example For example, if n = 5, and  = 1, and