1. More Control Charts Module 6

2. Why? There are many probability distributions in our world

4. The Binomial Distribution, The distribution of coin tosses.

7. Two types of data Variables----Continuous Attributes—Discrete, Countable –Two types of attributes data You can count occurrences and non-occurrences. You can only count occurences. Examples?????

8. Some Variables Shewart Charts X-mR aka i-Chart, Individuals Chart X-bar-range X-bar-sigma

9. Some Attributes Shewart Charts p-Chart np-Chart u-Chart c-Chart

10. Decide on type of data Continuo us (Variables ) Data Discrete (Attributes) Data More than one observation per subgroup? < 10 observations per subgroup? Can both occurrences & non- occurrences be counted? Are there equal area of opportunity ? Are the subgroup sizes equal? Ye s No –R–sXmRc-chart u-chartp-chartnp-chart

11. Example Individuals Chart

12. Example X-Range Chart

13. Example X-Sigma Chart

14. How did they do that?

15. The basic pattern…. Plot observed measurements over time. –Measurements, counts, rates Plot Centerline –Average measurement or count, pooled rate. Plot Control Limits –Centerline +/- Multiplier X “Standard Deviation”

16. Multiplier does 3 Things Determines the number of sigmas – usually 3 Converts standard deviations to standard errors (variables data). Can include factor to adjust for unusually small or large number of subgroups or time intervals. Note: How multiplier is constructed and used varies by author.

17. “Standard Deviation” Based on sample estimate of population standard deviation. Based on moving ranges. Based on ranges.

18. The i-Chart or XmR Chart Calculate average of all individual values = x Calculate all the moving ranges (MRi) MRi = |x i -x i-1 | Calculate the average MR = Rbar Calculate control limits = xbar +/- 2.66Rbar Plot xbar Plot control limits Plot individual values, points

19. The Xbar-Range Chart

20. The Xbar-Sigma Chart

21. The Xbar-Sigma Chart (Part II)

22. Is “3” always OK? Notice 3 is multiplied by the SD. This gives +/- 3 Sigma Control Limits. Designed for 25 observations. When you have only 7 observations –β risk is too high When you have 200 observations –α risk is too high Can use T-Sigma Limits

23. T-Sigma Limits No. of Plotted PointsT 21.5 3-42.0 5-92.5 10-343.0 35-1993.5 200-15004.0

24. How to use T-Sigma Limits Substitute the T-Sigma limit from the table for the “3” in A3, B3, and B4 above. For attributes charts, simply substitute the T-Sigma Limits for the multiplier in front of the standard error.

25. The attributes Shewart Charts p-Chart np-Chart u-Chart c-Chart

26. Example p-Chart

27. Example np-Chart

28. Example u-Chart

29. Example c-Chart

30. How did they do that?

31. The p-Chart

32. The np-Chart Pooled over all subgroups

33. The c-Chart

34. The u-Chart

35. Choosing Charts 1.Continuous A. Only 1 observation per subgroup—use iChart B. More than 1 observation/subgroup i) Less than 10 observations/subgroup—use Xbar-R ii) 10 or more observations/subgroup--use Xbar-Sigma 2.Attributes A. Occurrences (heads) and non-occurrences (tails) can be counted. i) Subgroups of equal size—use np-Chart Ii) Subgroups of unequal size—use p-Chart B. Only occurrences can be counted. i) Equal area of opportunity (denominators)—use c-Chart ii) Unequal area of opportunity– use u-Chart

36. Decide on type of data Continuous (Variables) Data Discrete (Attributes) Data More than one observation per subgroup? < 10 observations per subgroup? Can both occurrences & non-occurrences be counted? Are there equal area of opportunity? Are the subgroup sizes equal? Yes No –R–sXmRc-chart u-chartp-chartnp-chart See Flow Chart on page 72 of Carey and Lloyd