1/20/2016ENGM 720: Statistical Process Control1 ENGM Lecture 05 Variation Comparisons, Process Capability

1/20/2016 ENGM 720: Statistical Process Control 2 Assignment: Reading: Chapter 4 & 8 Finish reading through 4.3.4, , and CH 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

1/20/2016 ENGM 720: Statistical Process Control 3 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 Definitely NOT Different Probably NOT Different Probably Different

1/20/2016 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

1/20/2016 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), v 1 = n 1 – 1 degrees freedom for numerator, and v 2 = n 2 – 1 degrees freedom for the denominator Note:

1/20/2016 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 (  2 ) or should we estimate it by the sample variances (s 2 ) ? What are we trying to decide (alternate hypothesis)?

1/20/2016 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: F 1- , v1, v2 = 1/ F , v2, v1

1/20/2016 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 mean between the left and right sides of the training room Expectations? Direction of comparison? Significance Level? Known parameters? Best test? Result?

1/20/2016 ENGM 720: Statistical Process Control 9 Grip Strength Data Results R-L Side, Equal Variance Dominant Hand Means Comparison: L = x 1 = 129.4, S 1 2 = 2788, n 1 = 34 people R = x 2 = 104.0, S 2 2 = 1225, n 2 = 20 people S p = 47.1, v = 52 Two-Sided Test at  =.05 H A : There is a difference Test: Is | t 0 | > t.025, 52 ? |1.91| > NO! Keep the Null Hypothesis: There is NOT a difference btwn L & R !

1/20/2016 ENGM 720: Statistical Process Control 10 Grip Strength Data Results R-L Side, No Assumptions Dom. Hand Means Comparison: L = x 1 = 129.4, S 1 2 = 2788, n 1 = 34 people R = x 2 = 104.0, S 2 2 = 1225, n 2 = 20 people v = 51 Two-Sided Test at  =.05 H A : There is a difference Test: Is | t 0 | > t.025, 51 ? |2.12| > 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!

1/20/2016 ENGM 720: Statistical Process Control 11 Grip Strength Data Results Unknown σ 0 Variances Comparison: S 1 2 = 2788 n 1 = 34, v 1 = 33 S 2 2 = 1225 n 2 = 20, v 2 = 19 Two-Sided Test at  =.10 H A : There is a difference Test: Is F 0 > F.05, 33, 19 ? > YES! (Should also check F 1–  /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)

1/20/2016 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!

1/20/2016 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

1/20/2016 ENGM 720: Statistical Process Control 14 Process Capability - Timing Reduce Variability Identify Special Causes - Good (Incorporate) Improving Process Capability and Performance Characterize Stable Process Capability Head Off Shifts in Location, Spread Identify Special Causes - Bad (Remove) Continually Improve the System 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. Time Center the Process LSL  0 USL

1/20/2016 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

1/20/2016 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:

1/20/2016 ENGM 720: Statistical Process Control 17 Natural Tolerance Limits  +2  -2  +3  or UNTL -3  or LNTL ++ -- The Natural Tolerance Limits cover 99.73% of the process output   1  :68.26% of the total area   2  :95.46% of the total area   3  :99.73% of the total area

1/20/2016 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

1/20/2016 ENGM 720: Statistical Process Control 19 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) CumFreqCumFreq XXX CumFreqCumFreq CumFreqCumFreq

1/20/2016 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/d 2 (from Histogram) should be close to the standard deviation Can also estimate standard deviation by subtracting 50 th percentile from the 84 th percentile of the Histogram

1/20/2016 ENGM 720: Statistical Process Control 21 Process Capability Indices C p : Measures the potential capability of the current process - if the process were centered within the product specifications Two-sided Limits: One-sided Limit:

1/20/2016 ENGM 720: Statistical Process Control 22 C p Relation to Process Fallout Recommended Minimum Ratios: (D. C. Montgomery, 2001) Existing Process1.25 (1-sided)1.33 (2-sided) Existing, Safety / Critical Parameter New Process New, Safety / Critical Parameter

1/20/2016 ENGM 720: Statistical Process Control 23 Process Capability Indices C pk : Measures actual capability of current process - at its’ current location with respect to product specifications Formula: Where:

1/20/2016 ENGM 720: Statistical Process Control 24 Process Capability Indices Regarding C p and C pk : 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

1/20/2016 ENGM 720: Statistical Process Control 25 Process Capability Indices Two very different processes can have identical C pk values, though: because spread and location interact in C pk ! USL LSL

1/20/2016 ENGM 720: Statistical Process Control 26 Process Capability Indices C pm : 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:

1/20/2016 ENGM 720: Statistical Process Control 27 Process Capability Indices C pkm : Similar to C pm - just more sensitive to departures from the process target center point Not really in very common use Formula:

1/20/2016 ENGM 720: Statistical Process Control 28 Questions & Issues