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Stor 155, Section 2, Last Time

Stor 155, Section 2, Last Time. Distributions (how are data “spread out”?) Visual Display: Histograms Binwidth is critical Time Plots = Time Series Course Organization & Website http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155-07Home.html.

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Stor 155, Section 2, Last Time

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  1. Stor 155, Section 2, Last Time • Distributions (how are data “spread out”?) • Visual Display: Histograms • Binwidth is critical • Time Plots = Time Series • Course Organization & Website http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155-07Home.html

  2. Reading In Textbook Approximate Reading for Today’s Material: Pages 40-55 Approximate Reading for Next Class: Pages 64-83

  3. And now for something completely different Is this class too “monotone”? • Easier to understand? • Calm environment enhances learning? • Or does it induce somnolence? What is “somnolence”? Google definition: Sleepiness, a condition of semiconsciousness approaching coma.

  4. And now for something completely different An experiment: • Pull out any coins you have with you • How many of you have: • >= 1 penny? • >= 1 nickel? • >= 1 dime? • >= 1 quarter? • Choose most frequent denomination

  5. And now for something completely different Collect data (into Spreadsheet): • Years stamped on coins (chosen denomination) • Many as person has • Enter into spreadsheet • Look at “distribution” using histogram

  6. And now for something completely different • Predicted Answer • From Text Book, Problem 1.32 • Distribution is Left Skewed • Works out as predicted? • Why? • Note: most skewed dist’ns seem to be: Right Skewed

  7. Exploratory Data Analysis 4 Numerical Summaries of Quant. Variables: Idea: Summarize distributional information (“center”, “spread”, “skewed”) In Text, Sec. 1.2 for data (subscripts allow “indexing numbers” in list)

  8. Numerical Summaries • “Centers” (note there are several) • “Mean” = Average = • Greek letter “Sigma”, for “sum” In EXCEL, use “AVERAGE” function

  9. Numerical Summaries of Center • “Median” = Value in middle (of sorted list) Unsorted E.g: Sorted E.g: 3 0 1 1 27 “in middle”? (no) 2 better “middle”! 2 3 0 27 EXCEL: use function “MEDIAN”

  10. Difference Betw’n Mean & Median Symmetric Distribution: Essentially no difference Right Skewed Distribution: 50% area 50% area M bigger since “feels tails more strongly”

  11. Difference Betw’n Mean & Median Outliers (unusual values): Simple Web Example: http://www.stat.sc.edu/~west/applets/box.html • Mean feels outliers much more strongly • Leaves “range of most of data” • Good notion of “center”? (perhaps not) • Median affected very minimally • Robustness Terminology: Median is “resistant to the effect of outliers”

  12. Difference Betw’n Mean & Median A richer web example: Publisher’s Web Site: Statistical Applets: Mean & Median • For Symmetric distributions: • Both are same • Add an outlier: • Mean feels it much more strongly • Implication for “bad data”: can be very bad • Two Clusters: • Median jumps more quickly • Mean more stable (better?)

  13. Computation using Excel Some Toy Examples: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg3Done.xls • Compute Using Excel Functions • Mean feels location of data on number line • Median feels location of data in sorted list • Median breaks tie by averaging center points

  14. Numerical Centerpoint HW HW: 1.46 a, 1.47, 1.49 • Use EXCEL

  15. And now for something completely different Check out this small quick movie clip:

  16. And now for something completely different Suggestions for other things to show here are very welcome…. • Movie Clips… • Music… • Jokes… • Cartoons… • …

  17. Numerical Summaries (cont.) • “Spreads” (again there are several) 1. Range = biggest - smallest range Problems: • Feels only “outliers” • Not “bulk of data” • Very non-resistant to outliers

  18. Numerical Summaries of Spread • Variance = = “average squared distance to “ EXCEL: VAR Drawback: units are wrong e. g. For in feet  is in square feet

  19. Numerical Summaries of Spread • Standard Deviation EXCEL: STDEV • Scale is right • But not resistant to outliers • Will use quite a lot later (for reasons described later)

  20. Interactive View of S. D. Interesting web example (manipulate histogram): http://www.ruf.rice.edu/~lane/stat_sim/descriptive/index.html • Note SD range centered at mean • Can put SD “right near middle” (densely packed data) • Can put SD at “edges of data” (U shaped data) • Can put SD “outside of data” (big spike + outlier) • But generally “sensible measure of spread”

  21. Variance – S. D. HW C3: For the data set in 1.46 (i.e. 1.37), find the: • Variance (1620) • Standard Deviation (40.2) • Use EXCEL

  22. Numerical Summaries of Spread • Interquartile Range = IQR Based on “quartiles”, Q1 and Q3 (idea: shows where are 25% & 75% “through the data”) 25% 25% 25% 25% Q1 Q2 = median Q3 IQR = Q3 – Q1

  23. Quartiles Example Revisit Hidalgo Stamp Thickness example: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg6Done.xls Right skewness gives: • Median < Mean (mean “feels farther points more strongly”) • Q1 near median • Q3 quite far (makes sense from histogram)

  24. Quartiles Example A look under the hood: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg6Raw.xls • Can compute as separate functions for each • Or use: Tools  Data Analysis  Descriptive Stats • Which gives many other measures as well • Use “k-th largest & smallest” to get quartiles

  25. 5 Number Summary • Minimum • Q1 - 1st Quartile • Median • Q3 - 3rd Quartile • Maximum Summarize Information About: • Center - from 3 • Spread - from 2 & 4 (maybe 1 & 5) • Skewness - from 2, 3 & 4 • Outliers - from 1 & 5

  26. 5 Number Summary How to Compute? http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg6Done.xls • EXCEL function QUARTILE • “One stop shopping” • IQR seems to need explicit calculation

  27. Rule for Defining “Outliers” Caution: There are many of these Textbook version: Above Q3 + 1.5 * IQR Below Q1 – 1.5 * IQR For stamps data: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg6Done.xls • No outliers at “low end” • Some at “high end”

  28. 5 Number Sum. & Outliers HW 1.43

  29. Box Plot • Additional Visual Display Device • Again legacy from pencil & paper days • Not supported in EXCEL • So we won’t do • Main use: comparing populations • Example: Figure from text

  30. Box Plot

  31. Box Plot • Main use: comparing populations • Example: Figure from text • Want to do this? Find better software package than Excel

  32. And now for something completely different Recall Distribution of majors of students in this course:

  33. And now for something completely different How about a business manager joke? How many managers does it take to replace a light bulb?

  34. And now for something completely different How about a business manager joke? How many managers does it take to replace a light bulb? Two. One to find out if it needs changing, and one to tell an employee to change it. Source: http://www.joblatino.com/jokes/managers.html

  35. Linear Transformations Idea: What happens to data & summaries, when data are: “shifted and scaled” i.e. “panned and zoomed” Math: Scaled by a Shifted by b

  36. Linear Transformations Effect on linear summaries: • Centerpoints, and “follow data”: . • Spreads, and “feel scale, not shift”: .

  37. Most Useful Linear Transfo. “Standardization” Goal: put data sets on “common scale” Approach: • Subtract Mean , to “center at 0” • Divide by S.D. , to “give common SD = 1”

  38. Standardization Result is called “z-score”: Note that Thus is interpreted as: “number of SDs from the mean”

  39. Standardization Example Next time: work in Excel command: STANDARDIZE

  40. Standardization Example Buffalo Snowfall Data: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg7Done.xls • Standardized data have same (EXCEL default) histogram shape as raw data. (Since axes and bin edges just follow the transformation) • i.e. “shape” doesn’t depend on “scaling”

  41. Standardization Example A look under the hood: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg7Raw.xls Compute AVERAGE and SD • Standardize by: • Create Formula in cell B2 • Drag downwards • Keep Mean and SD cells fixed using $s 3. Check stand’d data have mean 0 & SD 1 note that “8.247E-16 = 0”

  42. Standardization HW C4: For data in 1.17, use EXCEL to: a. Give the list of standardized scores b. Give the Z-score for: (i) the mean (0) (ii) the median (-0.223) (iii) the smallest (-1.21) (iv) the largest (2.77) 1.59a, 1.73

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