1. Stat 155, Section 2, Last Time Numerical Summaries of Data: –Center: Mean, Medial –Spread: Range, Variance, S.D., IQR 5 Number Summary & Outlier Rule Transformation & Summaries 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 64-83 Approximate Reading for Next Class: Pages 102-112, 123-127

3. 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

4. And now for something completely different Unfortunately I lost the data… Didn’t save file??? Saved to Strange Location??? Anyway, I can’t find it… So won’t be able to finish this

5. A Special Request Professor Marron, I am having a lot of trouble creating time plots. Is there any way that you could walk me through creating one again or demonstrate on Tuesday? I read over the notes and the book but that didn't help. Thanks!

6. Exploratory Data Analysis 3 “Time Plots”, i.e. “Time Series: Idea: when time structure is important, plot variable as a function of time: variable time Often useful to “connect the dots”

7. Airline Passengers Example A look under the hood http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg5Done.xls Use Chart Wizard Chart Type: Line (or could do XY) Use subtype for points & lines Use menu for first log10 Although could just type it in Drag down to repeat for whole column

8. Modelling Distributions Text: Section 1.3 Idea: Approximate histograms by: an “idealized curve” i.e. a “density curve” that represents the underlying population

9. Idealized Curve Example Recall Hidalgo Stamps Data, Shifting Bin Movie (made # modes change): http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/StampsHistLoc.mpg Add idealized curve: http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/StampsHistLocKDE.mpg Note: “population curve” shows why histogram modes appear and disappear

10. Interpretation of Density Areas under density curve, give “relative frequency” Proportion of data between = = Area under =

11. Interpretation of Density Note: Total Area under density = 1 (since relative freq. of everything is 1) HW: 1.80 (a: l = w = 1 b: 0.25 c: 0.5), 1.81, 1.83 Work with pencil and paper, not EXCEL

12. Most Useful Density “Normal Curve” = “Gaussian Density” Shape: “like a mound” E.g. of “sand dumped from a truck” Older, worse, description: “bell shaped”

13. Normal Density Example Winter Daily Maximum Temperatures in Melbourne, Australia http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg8Done.xls Notes: Top Histogram is “mound shaped” Plus “small scale random variation” So model with “Normal Density”?

14. Normal Density Curves Note: there is a family of normal curves, indexed by: i.“Center”, i.e. Mean = ii.“Spread”, i.e. Stand. Deviation = Terminology: & are called “parameters” Greek “mu” ~ m Greek “sigma” ~ s

15. Family of Normal Curves Think about: “Shifts” (pans) indexed by “Scales” (zooms) indexed by Nice interactive graphical example: http://www.stat.sc.edu/~west/applets/normaldemo1.html (note area under curve is always 1)

16. Normal Curve Mathematics The “normal density curve” is: usual “function” of circle constant = 3.14… natural number = 2.7…

17. Normal Curve Mathematics Main Ideas: Basic shape is: “Shifted to mu”: “Scaled by sigma”: Make Total Area = 1: divide by as, but never

18. Normal Model Fitting Idea: Choose to give: “good” fit to data. Approach: IF the distribution is “mound shaped” & outliers are negligible THEN a “good” choice of normal model is:

19. Normal Fitting Example Revisit Melbourne Daily Max Temps http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg8Done.xls Fit curve, using “Visually good” approximation

20. Normal Fitting Example A look under the hood http://stat-or.unc.edu/webspace/postscript/marron/Teaching/stor155-2007/Stor155Eg8Done.xls Use chosen (not default) histogram bins for nice comparison bins Use longer range to avoid the “More” bin Can compute with density formula (Two steps, in cols F and G) Or use NORMDIST function (col J, check same as col G)

21. Normal Curve HW C5: A study of distance runners found a mean weight of 63.1 kg, with a standard deviation of 4.8 kg. Assuming that the distribution of weights is normal, use EXCEL to draw the density curve of the weight distribution.