1 / 18

IENG 486 - Lecture 05

IENG 486 - Lecture 05. Interpreting Variation Using Distributions. Assignment:. Reading: Chapter 1: (1.1, 1.3 – 1.4.5) Cursory – get Fig. 1.12., p.34; Deming Management,1.4.4 Liability Chapter 2: (2.2 – 2.7) Cursory – Define, Measure, Analyze, Improve, Control

asha
Download Presentation

IENG 486 - Lecture 05

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. IENG 486 - Lecture 05 Interpreting Variation Using Distributions IENG 486 Statistical Quality & Process Control

  2. Assignment: • Reading: • Chapter 1: (1.1, 1.3 – 1.4.5) • Cursory – get Fig. 1.12., p.34; Deming Management,1.4.4 Liability • Chapter 2: (2.2 – 2.7) • Cursory – Define, Measure, Analyze, Improve, Control • Chapter 3: (3.1, 3.3.1, 3.4.1) • HW 1: Chapter 3 Exercises: • 1, 3, 4 – using exam calculator • 10 (use Normal Plots spreadsheet from Materials page) • 43, 46, 47 (use Exam Tables from Materials page – Normal Dist.) IENG 486 Statistical Quality & Process Control

  3. Distributions • Distributions quantify the probability of an event • Events near the mean are most likely to occur, events further away are less likely to be observed 35.0  2.5 30.4 (-3) 34.8 (-) 39.2 (+) 43.6 (+3) 32.6 (-2) 37 () 41.4 (+2) IENG 486 Statistical Quality & Process Control

  4. Normal Distribution • Notation: r.v. • This is read: “x is normally distributed with mean m and standard deviation s.” • Standard Normal Distribution r.v. • (z represents a Standard Normal r.v.) IENG 486 Statistical Quality & Process Control

  5. Simple Interpretation of Standard Deviation of Normal Distribution IENG 486 Statistical Quality & Process Control

  6. Standard Normal Distribution • The Standard Normal Distribution has a mean () of 0 and a variance (2) of 1 (thus, standard deviation is also 1) • Total area under the curve, (z), from z = – to z =  is exactly 1 • The curve is symmetric about the mean • Half of the total area lays on either side, so: (– z) = 1 – (z) (z) z  IENG 486 Statistical Quality & Process Control

  7. Standard Normal Distribution • How likely is it that we would observe a data point more than 2.57 standard deviations beyond the mean? • Area under the curve from – to z = 2.57  is found by using the table on pp. 693-694, looking up the cumulative area for z = 2.57, and then subtracting the cumulative area from 1. (z) z  IENG 486 Statistical Quality & Process Control

  8. IENG 486 Statistical Quality & Process Control

  9. Standard Normal Distribution • How likely is it that we would observe a data point more than 2.57 standard deviations beyond the mean? • Area under the curve from – to z = 2.57  is found by using the table on pp. 693-694, looking up the cumulative area for z = 2.57, and then subtracting the cumulative area from 1. • Answer: 1 – .99492 = .00508, or about 5 times in 1000 (z) z  IENG 486 Statistical Quality & Process Control

  10. What if the distribution isn’t a Standard Normal Distribution? • If it is from anyNormal Distribution, we can express the difference from a sample mean to the population mean in units of the standard deviation, and this converts it to a Standard Normal Distribution. • Conversion formula is: where: x is the sample location point,  is the population mean, and  is the population standard deviation. IENG 486 Statistical Quality & Process Control

  11. What if the distribution isn’t even a Normal Distribution? • The Central Limit Theorem allows us to take the sum of several means, regardless of their distribution, and approximate this sum using the Normal Distribution if the number of observations is large enough. • Most assemblies are the result of adding together components, so if we take the sum of the means for each component as an estimate for the entire assembly, we meet the CLT criteria. • If we take the mean of a sample from a distribution, we meet the CLT criteria (think of how the mean is computed). IENG 486 Statistical Quality & Process Control

  12. Example: Process Yield • Specifications are often set irrespective of process distribution, but if we understand our process we can estimate yield / defects. • Assume a specification calls for a value of 35.0  2.5. • Assume the process has a distribution that is Normally distributed, with a mean of 37.0 and a standard deviation of 2.20. • Estimate the proportion of the process output that will meet specifications. IENG 486 Statistical Quality & Process Control

  13. Continuous Probability of a range of outcomes is the area under the PDF (integration) Discrete Probability of a range of outcomes is the area under the PDF (sum discrete outcomes) 35.0  2.5 35.0  2.5 () 30.4 (-3) 34.8 (-) 39.2 (+) 43.6 (+3) 30 34 38 42 32.6 (-2) 37 () 41.4 (+2) 32 36 40 Continuous & Discrete Distributions IENG 486 Statistical Quality & Process Control

  14. Discrete Distribution Example • Sum of two six-sided dice: • Outcomes range from 2 to 12. • Count the possible ways to obtain each individual sum - forms a histogram • What is the most frequently occurring sum that you could roll? • Most likely outcome is a sum of 7 (there are 6 ways to obtain it) • What is the probability of obtaining the most likely sum in a single roll of the dice? • 6  36 = .167 • What is the probability of obtaining a sum greater than 2 and less than 11? • 32  36 = .889 IENG 486 Statistical Quality & Process Control

  15. How do we know what the distribution is when all we have is a sample? • Theory – “CLT applies to measurements taken consisting of many assemblies…” • Experience – “past use of a distribution has generated very good results…” • “Testing” – combination of the above … in this case, anyway! • If we know the generating function for a distribution, we can construct a grid (probability paper) that will allow us to observe a straight line when sufficient data from that distribution are plotted on the grid • Easiest grid to create is the Standard Normal Distribution … • because it is an easy transformation to “standard“ parameters IENG 486 Statistical Quality & Process Control

  16. Normal Probability Plots • Take raw data and count observations (n) • Set up a column of j values (1 to j) • Compute F(zj) for each j value • F(zj) = (j - 0.5)/n • Get zj value for each F(zj) in Standard Normal Table • Find table entry(F(zj)), then read index value (zj) • Set up a column of sorted, observed data • Sorted in increasing value • Plot zj values versus sorted data values • Approximate with sketched line at 25% and 75% points IENG 486 Statistical Quality & Process Control

  17. Interpreting Normal Plots • Assess Equal-Variance and Normality assumptions • Data from a Normal sample should tend to fall along the line, so if a “fat pencil” covers almost all of the points, then a normality assumption is supported • The slope of the line reflects the variance of the sample, so equal slopes support the equal variance assumption • Theoretically: • Sketched line should intercept the zj = 0 axis at the mean value • Practically: • Close is good enough for comparing means • Closer is better for comparing variances • If the slopes differ much for two samples, use a test that assumes the variances are not the same IENG 486 Statistical Quality & Process Control

  18. Normal Probability Plots Tools for constructing Normal Probability Plots: • (Normal) Probability Paper • In-class handout • Normal Plots Template • Materials Page on course website • Interpretation • Fat Pencil Test IENG 486 Statistical Quality & Process Control

More Related