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**What is the point of these sports?**

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… Who is the better shot? Shot a rifle? Played darts?**

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

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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 8 7 10 9 Deviation = distance between observations and the mean (or average) Emmett Observations 10 9 8 7 averages 8.4 Deviations = 1.6 9 – 8.4 = 0.6 8 – 8.4 = -0.4 7 – 8.4 = -1.4 0.0 Jake

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Variability Deviation = distance between observations and the mean (or average) Emmett Observations 7 6 averages 6.6 Deviations 7 – 6.6 = 0.4 6 – 6.6 = -0.6 0.0 7 6 Jake

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Variability 8 7 10 9 Variance = average distance between observations and the mean squared Emmett Observations 10 9 8 7 averages 8.4 Deviations = 1.6 9 – 8.4 = 0.6 8 – 8.4 = -0.4 7 – 8.4 = -1.4 0.0 Squared Deviations 2.56 0.36 0.16 1.96 1.0 Jake Variance

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Variability Variance = average distance between observations and the mean squared Emmett Observations 7 6 averages Deviations Squared Deviations 7 6 Jake

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Variability Variance = average distance between observations and the mean squared Emmett Observations 7 6 averages 6.6 Deviations = 0.4 6 – 6.6 = -0.6 0.0 Squared Deviations 0.16 0.36 0.24 7 6 Jake Variance

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**But what good is a standard deviation**

Variability Standard deviation = square root of variance Emmett Variance Standard Deviation Emmett 1.0 Jake 0.24 Jake But what good is a standard deviation ?

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**Variability The world tends to be bell-shaped Even very rare**

outcomes are possible (probability > 0) Fewer in the “tails” (lower) (upper) Most outcomes occur in the middle

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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? One observation falls outside 3 standard deviations?**

One observation falls in zone A? 2 out of 3 observations fall in one zone A? 2 out of 3 observations fall in one zone B or beyond? 4 out of 5 observations fall in one zone B or beyond? 8 consecutive points above the mean, rising, or falling? X X XX X X X X XX

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**SPC uses samples to identify that special causes have occurred**

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: Specification 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**

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

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**Usual v. Unusual, Acceptable v. Defective**

μ Target

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**Cpk measures “Process Capability”**

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

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**Process capability Good quality: defects are rare (Cpk>1)**

Poor quality: defects are common (Cpk<1) = USL – x 3σ 24 – 20 3(2) = = .667 Cpk = min = x - LSL 3σ 20 – 15 3(2) = = .833 = = 3σ = (UPL – x, or x – LPL)

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Going out of control When an observation is unusual, what can we conclude? μ2 The mean has changed X μ1

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Going out of control When an observation is unusual, what can we conclude? The standard deviation has changed σ2 σ1 X

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**Setting up control charts: Calculating the limits**

Sample n items (often 4 or 5) Find the mean of the sample (x-bar) Find the range of the sample R Plot on the chart Plot the R on an R chart Repeat steps 1-5 thirty times Average the ’s to create (x-bar-bar) Average the R’s to create (R-bar)

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**Setting up control charts: Calculating the limits**

Find A2 on table (A2 times R estimates 3σ) Use formula to find limits for x-bar chart: Use formulas to find limits for R chart:

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**Let’s try a small problem**

smpl 1 smpl 2 smpl 3 smpl 4 smpl 5 smpl 6 observation 1 7 11 6 10 observation 2 8 5 observation 3 12 x-bar R X-bar chart R chart UCL Centerline LCL

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**Let’s try a small problem**

smpl 1 smpl 2 smpl 3 smpl 4 smpl 5 smpl 6 Avg. observation 1 7 11 6 10 observation 2 8 5 observation 3 12 X-bar 7.3333 9.6667 9.3333 7.6667 8.0556 R 1 3 3.5 X-bar chart R chart UCL 9.0125 Centerline 8.0556 3.5 LCL 4.4751

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X-bar chart 8.0556 4.4751

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R chart 9.0125 3.5

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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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**Show real time examples of charts here**

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

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**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 “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 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)

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Probability Distributions CSLU 2850.Lo1 Spring 2008 Cameron McInally Fordham University May contain work from the Creative Commons.

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