More Control Charts

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 Continuous (Variables) Data Discrete (Attributes) Data Can both occurrences & non-occurrences be counted? Yes More than one observation per subgroup? No No Yes < 10 observations per subgroup? Are there equal area of opportunity? Are the subgroup sizes equal? Yes No Yes No No Yes –R –s XmR c-chart u-chart p-chart np-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 = |xi-xi-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

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