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Published byAshley McClure Modified over 3 years ago

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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?

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Have you ever… Shot a rifle? Played darts? Shot a round of golf? Played basketball? Emmett Jake Who is the better shot?

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Discussion What do you measure in your process? Why do those measures matter? Are those measures consistently the same? Why not?

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Variability Deviation = distance between observations and the mean (or average) Emmett Jake Observations averages8.4 Deviations = – 8.4 = – 8.4 = – 8.4 =

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Variability Deviation = distance between observations and the mean (or average) Emmett Jake Observations averages6.6 Deviations 7 – 6.6 = – 6.6 =

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Variability Variance = average distance between observations and the mean squared Emmett Jake Observations averages8.4 Deviations = – 8.4 = – 8.4 = – 8.4 = Squared Deviations Variance

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Variability Variance = average distance between observations and the mean squared Emmett Jake Observations averages DeviationsSquared Deviations

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Variability Variance = average distance between observations and the mean squared Emmett Jake Observations averages6.6 Deviations = – 6.6 = Squared Deviations Variance

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Variability Standard deviation = square root of variance Emmett Jake VarianceStandard Deviation Emmett1.0 Jake But what good is a standard deviation

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Variability The world tends to be bell-shaped Most outcomes occur in the middle Fewer in the tails (lower) Fewer in the tails (upper) Even very rare outcomes are possible (probability > 0) Even very rare outcomes are possible (probability > 0)

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Variability Here is why: Even outcomes that are equally likely (like dice), when you add them up, become bell shaped

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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

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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 X XX X X X XX

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Causes of Variability Common Causes: Random variation (usual) No pattern Inherent in process adjusting the process increases its variation Special Causes Non-random variation (unusual) May exhibit a pattern Assignable, explainable, controllable adjusting the process decreases its variation SPC uses samples to identify that special causes have occurred

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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 Limits = target ± tolerance Define acceptable & unacceptable

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Process vs. control limits Variance of averages < variance of individual items Distribution of averages Control limits Process limits Distribution of individuals Specification limits

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Usual v. Unusual, Acceptable v. Defective μ Target A B C D E

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More about limits Good quality: defects are rare (C pk >1) Poor quality: defects are common (C pk <1) C pk measures Process Capability If process limits and control limits are at the same location, C pk = 1. C pk 2 is exceptional. μ target μ target

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Process capability Good quality: defects are rare (Cpk>1) Poor quality: defects are common (Cpk<1) Cpk = min USL – x 3σ = x - LSL 3σ = 3σ = (UPL – x, or x – LPL) = = – 20 3(2) = = – 15 3(2) = =.833

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Going out of control When an observation is unusual, what can we conclude? μ2μ2 The mean has changed X μ1μ1

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Going out of control When an observation is unusual, what can we conclude? The standard deviation has changed σ2σ2 X σ1σ1

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Setting up control charts: Calculating the limits 1. Sample n items (often 4 or 5) 2. Find the mean of the sample (x-bar) 3. Find the range of the sample R 4. Plot on the chart 5. Plot the R on an R chart 6. Repeat steps 1-5 thirty times 7. Average the s to create (x-bar-bar) 8. Average the Rs to create (R-bar)

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Setting up control charts: Calculating the limits 9. Find A 2 on table (A 2 times R estimates 3σ) 10. Use formula to find limits for x-bar chart: 11. Use formulas to find limits for R chart:

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Lets try a small problem smpl 1smpl 2smpl 3smpl 4smpl 5smpl 6 observation observation observation x-bar R X-bar chartR chart UCL Centerline LCL

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Lets try a small problem smpl 1smpl 2smpl 3smpl 4smpl 5smpl 6Avg. observation observation observation X-bar R X-bar chartR chart UCL Centerline LCL

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X-bar chart

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R chart

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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-of- control Random causes will on rare occasions indicate the process is probably out-of- control when it actually is not

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Interpreting charts In the excel spreadsheet, look for these shifts: AB C D Show real time examples of charts here

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Lots of other charts exist P chartC chartsU chartsCusum & 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 V shaped or Curved control limits (calculate them by hiring a statistician)

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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

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Chart advice Larger samples are more accurate Sample costs money, but so does being out-of-control Dont convert measurement data to yes/no binomial data (Xs to Ps) Not all out-of control points are bad Dont combine data (or mix product) Have out-of-control procedures (what do I do now?) Actual production volume matters (Average Run Length)

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