ENGM 720 - Lecture 05

1. ENGM 720 - Lecture 05 Variation Comparisons, Process Capability ENGM 720: Statistical Process Control

2. Assignment: • Reading: • Chapter 4 & 8 • Finish reading through 4.3.4, 4.4 - 4.4.3, and CH 8 - 8.3 • Begin reading 4.5 • Chapter 5 • Begin reading through 5.2, and 5.4 • Assignments: • Obtain the Hypothesis Test (Chart &) Tables – Materials Page • Obtain the Exam Tables DRAFT– Materials Page • Verify accuracy as you work assignments • Access New Assignment and Previous Assignment Solutions: • Download Assignment 3 Instruction & Solutions ENGM 720: Statistical Process Control

3. Probably Different Probably NOT Different Definitely NOT Different Comparison of Variances • The second types of comparison are those that compare the spread of two distributions. To do this: • Compute the ratio of the two variances, and then compare the ratio to one of two known distributions as a check to see if the magnitude of that ratio is sufficiently unlikely for the distribution. • The assumption that the data come from Normal distributions is very important. Assess how normally data are distributed prior to conducting either test. Definitely Different ENGM 720: Statistical Process Control

4. Situation VII: Variance Test With 0 Known • Used when: • existing comparison process has been operating without much change in variation for a long time • Procedure: • form ratio of a sample variance (t-distribution variable) to a population variance (Normal distribution variable), v = n - 1 degrees of freedom ENGM 720: Statistical Process Control

5. Situation VIII: Variance Test With 0 Unknown • Use: • worst case variation comparison process for when there is not enough prior history • Procedure: • form ratio of the sample variances (two 2-distributions), v1 = n1 – 1 degrees freedom for numerator, and v2 = n2 – 1 degrees freedom for the denominator Note: ENGM 720: Statistical Process Control

6. Table for Variance Comparisons • Decision on which test to use is based on answering the following: • Do we know a theoretical variance (s2) or should we estimate it by the sample variances (s2) ? • What are we trying to decide (alternate hypothesis)? ENGM 720: Statistical Process Control

7. Table for Variance Comparisons • These questions tell us: • What sampling distribution to use • What test statistic(s) to use • What criteria to use • How to construct the confidence interval • Four primary test statistics for variance comparisons • Two sampling distributions • Two confidence intervals • Six alternate hypotheses • Table construction • Note: F1-a, v1, v2 = 1/ Fa, v2, v1 ENGM 720: Statistical Process Control

8. Grip Strength Example • True Corporate Training Example • How could grip strength vary among people in the SPC training room? • Data collection to detect difference in dominant hand meanbetween the left and right sides of the training room • Expectations? • Direction of comparison? • Significance Level? • Known parameters? • Best test? • Result? ENGM 720: Statistical Process Control

9. R-L Side, Equal Variance Dominant Hand Means Comparison: L = x1 = 129.4, S12 = 2788, n1 = 34 people R = x2 = 104.0, S22 = 1225, n2 = 20 people Sp = 47.1, v = 52 Two-Sided Test at  = .05 HA: There is a difference Test: Is | t0 | > t.025, 52? |1.91| > 2.009 - NO! Keep the Null Hypothesis: There is NOT a difference btwn L & R ! Grip Strength Data Results ENGM 720: Statistical Process Control

10. R-L Side, No Assumptions Dom. Hand Means Comparison: L = x1 = 129.4, S12 = 2788, n1 = 34 people R = x2 = 104.0, S22 = 1225, n2 = 20 people v = 51 Two-Sided Test at  = .05 HA: There is a difference Test: Is | t0 | > t.025, 51? |2.12| > 2.009 - YES! Reject the Null Hypothesis: There IS a difference btwn L & R! Why is this wimpy test significant when the other wasn’t? ANS: Check the equal variance assumption! Grip Strength Data Results ENGM 720: Statistical Process Control

11. Unknown σ0 Variances Comparison: S12 = 2788 n1 = 34, v1 = 33 S22 = 1225 n2 = 20, v2 = 19 Two-Sided Test at  = .10 HA: There is a difference Test: Is F0 > F.05, 33, 19? 2.276 > 2.07 - YES! (Should also checkF1– /2, 33, 19) Reject the Null Hypothesis: There IS a difference in variance! At  = .05, this test is just barely not significant (Should also have checked for Normality with Normal Prob. Plot) Grip Strength Data Results ENGM 720: Statistical Process Control

12. Statistical Quality Improvement • Goal: Control and Reduction of Variation • Causes of Variation: • Chance Causes / Common Causes • In Statistical Control • Natural variation / background noise • Assignable Causes / Special Causes • Out of Statistical Control • Things we can do something about - IF we act quickly! • Both can cause defects – because specifications are often set regardless of process capabilities! ENGM 720: Statistical Process Control

13. Process Capability • Process Capability Analysis (PCA) • Is only done when the process is in a state of Statistical Control • Meaning: NO SPECIAL CAUSES are present • Process does not have to be centered to do PCA • Yield will improve if process is centered, but the value is in knowing what / where to improve the process • PCA is done periodically when the process has been operating in a state of statistical control • Allows for measuring improvement over time • Allows for marketing your competitive edge ENGM 720: Statistical Process Control

14. Process Capability Analysis is performed when there are NO special causes of variability present – ie. when the process is in a state of statistical control, as illustrated at this point. Improving Process Capability and Performance Continually Improve the System Characterize Stable Process Capability Head Off Shifts in Location, Spread Time Identify Special Causes - Bad (Remove) Identify Special Causes - Good (Incorporate) Reduce Variability Center the Process LSL 0 USL Process Capability - Timing ENGM 720: Statistical Process Control

15. Process Capability • Process Capability is INDEPENDENT of product specifications • Most specifications are set without regard for process capability • However, understanding process capability helps the engineer to set more reasonable specifications • PCA reflects only the Natural Tolerance Limits of the process • PCA is done by examining the process • Histogram • Normal Probability Plot ENGM 720: Statistical Process Control

16. Natural Tolerance Limits • The natural tolerance limits assume: • The process is well-modeled by the Normal Distribution • Three sigma is an acceptable proportion of the process to yield • The Upper and Lower Natural Tolerance Limits are derived from: • The process mean () and • The process standard deviation () • Equations: ENGM 720: Statistical Process Control

17.  1 :68.26% of the total area  2 :95.46% of the total area  3 :99.73% of the total area -3 or LNTL - + +3 or UNTL -2  +2 The Natural Tolerance Limits cover 99.73% of the process output Natural Tolerance Limits ENGM 720: Statistical Process Control

18. PCA: Histogram Construction • Verify rough shape and location of histogram • Symmetric (roughly bell-shaped) • Mean  median  mode • Quickly confirm applicability prior to statistical analysis • Often hard to distinguish a Normal Distribution from a t-Distribution • Sometimes even a Normal distribution doesn’t look normal • More data and columns (bins) can make a difference • Verify location of process with respect to Specifications • Quick inspection will show what to do to improve the process ENGM 720: Statistical Process Control

19. C u m F r e q C u m F r e q C u m F r e q X X X PCA: Normal Probability Plot • A Normal Plot better clarifies whether the distribution is Normal by a visual inspection for: • Non-random patterns (non-Normal) • Fat Pencil Test (Normal if passes) ENGM 720: Statistical Process Control

20. PCA: Parameter Estimation • The Normal Plot mid-point estimates the process mean • The slope of the “best fit” line for the Normal Plot estimates the standard deviation • Choose the 25th and 75th percentile points to calculate the slope • The Histogram mode should be close to the mean • The range/d2 (from Histogram) should be close to the standard deviation • Can also estimate standard deviation by subtracting 50th percentile from the 84th percentile of the Histogram ENGM 720: Statistical Process Control

21. Process Capability Indices • Cp: • Measures the potential capability of the current process - if the process were centered within the product specifications • Two-sided Limits: • One-sided Limit: ENGM 720: Statistical Process Control

22. Cp Relation to Process Fallout • Recommended Minimum Ratios: (D. C. Montgomery, 2001) • Existing Process 1.25 (1-sided) 1.33 (2-sided) • Existing, Safety / Critical Parameter 1.45 1.50 • New Process 1.45 1.50 • New, Safety / Critical Parameter 1.60 1.67 ENGM 720: Statistical Process Control

23. Process Capability Indices • Cpk: • Measures actual capability of current process - at its’ current location with respect to product specifications • Formula: Where: ENGM 720: Statistical Process Control

24. Process Capability Indices • Regarding Cp and Cpk: • Both assume that the process is Normally distributed • Both assume that the process is in Statistical Control • When they are equal to each other, the process is perfectly centered • Both are pretty common reporting ratios among vendors and purchasers ENGM 720: Statistical Process Control

25. LSL USL Process Capability Indices • Two very different processes can have identical Cpk values, though: • because spread and location interact in Cpk! ENGM 720: Statistical Process Control

26. Process Capability Indices • Cpm: • Measures the current capability of the process - using the process target center point within the product specifications in the calculation • Formula: Where target T is: ENGM 720: Statistical Process Control

27. Process Capability Indices • Cpkm: • Similar to Cpm - just more sensitive to departures from the process target center point • Not really in very common use • Formula: ENGM 720: Statistical Process Control

28. Questions & Issues ENGM 720: Statistical Process Control