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SPC Basic Dr. Mohamed Riyazh Khan – DoMS SNS. College of Engineering Have you ever… Shot a rifle? Played darts? Played basketball? Shot a round of golf? What is the point of these sports? What makes them hard? Have you ever… Shot a rifle? Played darts? Shot a round of golf? Played basketball? Emmett Jake Who is the better shot? Discussion What do you measure in your process? Why do those measures matter? Are those measures consistently the same? Why not? Variability Deviation = distance between observations and the mean (or average) Observations 8 7 10 8 9 Emmett Deviations 10 10 - 8.4 = 1.6 9 9 – 8.4 = 0.6 8 8 – 8.4 = -0.4 8 8 – 8.4 = -0.4 7 7 – 8.4 = -1.4 averages 8.4 0.0 Jake Variability Deviation = distance between observations and the mean (or average) Observations Deviations 7 7 – 6.6 = 0.4 7 7 – 6.6 = 0.4 7 7 – 6.6 = 0.4 6 6 – 6.6 = -0.6 6 6 – 6.6 = -0.6 averages 6.6 0.0 Emmett 7 6 7 7 6 Jake Variability 8 7 10 8 9 Variance = average distance between observations and the mean squared Deviations Squared Deviations 10 10 - 8.4 = 1.6 2.56 9 – 8.4 = 0.6 0.36 8 8 – 8.4 = -0.4 0.16 8 8 – 8.4 = -0.4 0.16 7 7 – 8.4 = -1.4 1.96 0.0 1.0 Observations 9 averages Emmett 8.4 Jake Variance Variability Variance = average distance between observations and the mean squared Observations 7 7 7 6 6 averages Deviations Squared Deviations Emmett 7 6 7 7 6 Jake Variability Variance = average distance between observations and the mean squared Deviations Squared Deviations 7 7 - 6.6 = 0.4 0.16 7 7 - 6.6 = 0.4 0.16 7 7 - 6.6 = 0.4 0.16 6 6 – 6.6 = -0.6 0.36 6 6 – 6.6 = -0.6 0.36 0.0 0.24 Observations averages 6.6 Emmett 7 6 7 7 6 Jake Variance Variability Standard deviation = square root of variance Variance Emmett Jake 1.0 0.24 Standard Deviation 1.0 0.4898979 But what good is a standard deviation Emmett Jake Variability The world tends to be bell-shaped Even very rare outcomes are possible (probability > 0) Fewer in the “tails” (lower) Most outcomes occur in the middle Fewer in the “tails” (upper) Even very rare outcomes are possible (probability > 0) Variability Here is why: Even outcomes that are equally likely (like dice), when you add them up, become bell shaped Add up the dots on the dice Probability 0.2 0.15 1 die 0.1 2 dice 0.05 3 dice 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Sum of dots “Normal” bell shaped curve Add up about 30 of most things and you start to be “normal” Normal distributions are divide up into 3 standard deviations on each side of the mean Once your that, you know a lot about what is going on And that is what a standard deviation is good for Usual or unusual? 1. One observation falls outside 3 standard deviations? 2. One observation falls in zone A? 3. 2 out of 3 observations fall in one zone A? 4. 2 out of 3 observations fall in one zone B or beyond? 5. 4 out of 5 observations fall in one zone B or beyond? 6. 8 consecutive points above the mean, rising, or falling? X XX X 34 56 X1X XX2 X 78 Causes of Variability Common Causes: Random variation (usual) No pattern Inherent in process adjusting the process increases its variation Special Causes Non-random variation May exhibit a pattern (unusual) Assignable, explainable, controllable adjusting the process decreases its variation SPC uses samples to identify that special causes have occurred Limits Process and Control limits: Statistical Process limits are used for individual items Control limits are used with averages Limits = μ ± 3σ Define usual (common causes) & unusual (special causes) Specification limits: Engineered = target ± tolerance Define acceptable & unacceptable Limits Process vs. control limits Distribution of averages Control limits Specification limits Variance of averages < variance of individual items Distribution of individuals Process limits Usual v. Unusual, Acceptable v. Defective A B C μ Target D E More about limits Good quality: defects are rare (Cpk>1) μ target μ target Poor quality: defects are common (Cpk<1) Cpk measures “Process Capability” If process limits and control limits are at the same location, Cpk = 1. Cpk ≥ 2 is exceptional. Process capability Good quality: defects are rare (Cpk>1) Poor quality: defects are common (Cpk<1) Cpk = min = USL – x = 24 – 20 =.667 3σ 3(2) = x - LSL = 20 – 15 =.833 3σ 3(2) 14 = = 3σ = (UPL – x, or x – LPL) 15 20 24 26 Going out of control When an observation is unusual, what can we conclude? The mean has changed X μ1 μ2 Going out of control When an observation is unusual, what can we conclude? σ1 The standard deviation has changed σ2 X Setting up control charts: Calculating the limits 1. 2. 3. 4. 5. 6. 7. 8. Sample n items (often 4 or 5) Find the mean of the sample x (x-bar) Find the range of the sample R Plot x on the x chart Plot the R on an R chart Repeat steps 1-5 thirty times Average the x ’s to create x (x-bar-bar) Average the R’s to create R (R-bar) Setting up control charts: Calculating the limits 9. 10. Find A2 on table (A2 times R estimates 3σ) Use formula to find limits for x-bar chart: X A2 R 11. Use formulas to find limits for R chart: LCL D3 R UCL D4 R Let’s try a small problem smpl 1 smpl 2 smpl 3 smpl 4 smpl 5 smpl 6 observation 1 7 11 6 7 10 10 observation 2 7 8 10 8 5 5 observation 3 8 10 12 7 6 8 x-bar R X-bar chart UCL Centerline LCL R chart Let’s try a small problem smpl 1 smpl 2 smpl 3 smpl 4 smpl 5 smpl 6 observation 1 7 11 6 7 10 10 observation 2 7 8 10 8 5 5 observation 3 8 10 12 7 6 8 X-bar 7.3333 9.6667 9.3333 7.3333 7 7.6667 R 1 3 6 1 5 5 X-bar chart 8.0556 3.5 R chart 11.6361 9.0125 Centerline 8.0556 3.5 LCL 4.4751 0 UCL Avg. X-bar chart 14.0000 12.0000 10.0000 8.0000 11.6361 8.0556 6.0000 4.0000 2.0000 0.0000 4.4751 1 2 3 4 5 6 R chart 10 9.0125 8 6 4 3.5 2 0 0 1 2 3 4 5 6 Interpreting charts Observations outside control limits indicate the process is probably “out-of-control” Significant patterns in the observations indicate the process is probably “out-ofcontrol” Random causes will on rare occasions indicate the process is probably “out-ofcontrol” when it actually is not Interpreting charts In the excel spreadsheet, look for these shifts: A B C Show real time examples of charts here D Lots of other charts exist P chart C charts U charts Cusum & EWMA For yes-no questions like “is it defective?” (binomial data) For counting number defects where most items have ≥1 defects (eg. custom built houses) Average count per unit (similar to C chart) Advanced charts p(1 p) p 3 n c 3 c u u 3 n “V” shaped or Curved control limits (calculate them by hiring a statistician) Selecting rational samples Chosen so that variation within the sample is considered to be from common causes Special causes should only occur between samples Special causes to avoid in sampling passage of time workers shifts machines Locations Chart advice Larger samples are more accurate Sample costs money, but so does being out-of-control Don’t convert measurement data to “yes/no” binomial data (X’s to P’s) Not all out-of control points are bad Don’t combine data (or mix product) Have out-of-control procedures (what do I do now?) Actual production volume matters (Average Run Length)