1. Lecture 3 Dustin Lueker

2.  Simple Random Sampling (SRS) ◦ Each possible sample has the same probability of being selected  Stratified Random Sampling ◦ The population can be divided into a set of non-overlapping subgroups (the strata) ◦ SRSs are drawn from each strata  Cluster Sampling ◦ The population can be divided into a set of non-overlapping subgroups (the clusters) ◦ The clusters are then selected at random, and all individuals in the selected clusters are included in the sample  Systematic Sampling ◦ Useful when the population consists as a list ◦ A value K is specified. Then one of the first K individuals is selected at random, after which every Kth observation is included in the sample STA 291 Summer 2010 Lecture 32

3. 3  Summarize data ◦ Condense the information from the dataset  Graphs  Table  Numbers  Interval data ◦ Histogram  Nominal/Ordinal data ◦ Bar chart ◦ Pie chart

4.  Difficult to see the “big picture” from these numbers ◦ We want to try to condense the data STA 291 Summer 2010 Lecture 34 Alabama 11.6Alaska 9.0 Arizona 8.6Arkansas 10.2 California 13.1Colorado 5.8 Connecticut 6.3Delaware 5.0 D C 78.5Florida 8.9 Georgia 11.4Hawaii 3.8 ……

5. STA 291 Summer 2010 Lecture 35  A listing of intervals of possible values for a variable  Together with a tabulation of the number of observations in each interval.

6. STA 291 Summer 2010 Lecture 36 Murder RateFrequency 0-2.95 3-5.916 6-8.912 9-11.912 12-14.94 15-17.90 18-20.91 >211 Total51

7.  Conditions for intervals ◦ Equal length ◦ Mutually exclusive  Any observation can only fall into one interval ◦ Collectively exhaustive  All observations fall into an interval  Rule of thumb: ◦ If you have n observations then the number of intervals should approximately STA 291 Summer 2010 Lecture 37

8. 8  Relative frequency for an interval ◦ Proportion of sample observations that fall in that interval  Sometimes percentages are preferred to relative frequencies

9. STA 291 Summer 2010 Lecture 39 Murder RateFrequencyRelative Frequency Percentage 0-2.95.1010 3-5.916.3131 6-8.912.2424 9-11.912.2424 12-14.94.088 15-17.9000 18-20.91.022 >211.022 Total511100

10. STA 291 Summer 2010 Lecture 310  Notice that we had to group the observations into intervals because the variable is measured on a continuous scale ◦ For discrete data, grouping may not be necessary  Except when there are many categories  Intervals are sometimes called classes ◦ Class Cumulative Frequency  Number of observations that fall in the class and in smaller classes ◦ Class Relative Cumulative Frequency  Proportion of observations that fall in the class and in smaller classes

11. STA 291 Summer 2010 Lecture 311 Murder RateFrequencyRelative Frequency Cumulative Frequency Relative Cumulative Frequency 0-2.95.105 3-5.916.3121.41 6-8.912.2433.65 9-11.912.2445.89 12-14.94.0849.97 15-17.90049.97 18-20.91.0250.99 >211.02511 Total511 1

12. STA 291 Summer 2010 Lecture 312  Use the numbers from the frequency distribution to create a graph ◦ Draw a bar over each interval, the height of the bar represents the relative frequency for that interval ◦ Bars should be touching  Equally extend the width of the bar at the upper and lower limits so that the bars are touching.

13. STA 291 Summer 2010 Lecture 313

14. STA 291 Summer 2010 Lecture 314

15. STA 291 Summer 2010 Lecture 315  Histogram: for interval (quantitative) data  Bar graph is almost the same, but for qualitative data  Difference: ◦ The bars are usually separated to emphasize that the variable is categorical rather than quantitative ◦ For nominal variables (no natural ordering), order the bars by frequency, except possibly for a category “other” that is always last

16.  First Step ◦ Create a frequency distribution STA 291 Summer 2010 Lecture 316 Highest Degree ObtainedFrequency (Number of Employees) Grade School15 High School200 Bachelor’s185 Master’s55 Doctorate70 Other25 Total550

17.  Bar graph ◦ If the data is ordinal, classes are presented in the natural ordering STA 291 Summer 2010 Lecture 317

18.  Pie is divided into slices ◦ Area of each slice is proportional to the frequency of each class STA 291 Summer 2010 Lecture 318

19. STA 291 Summer 2010 Lecture 319

20. 20  Write the observations ordered from smallest to largest ◦ Looks like a histogram sideways ◦ Contains more information than a histogram, because every single observation can be recovered  Each observation represented by a stem and leaf  Stem = leading digit(s)  Leaf = final digit STA 291 Summer 2010 Lecture 320

21. 21STA 291 Summer 2010 Lecture 321 Stem Leaf # 20 3 1 19 18 17 16 15 14 13 135 3 12 7 1 11 334469 6 10 2234 4 9 08 2 8 03469 5 7 5 1 6 034689 6 5 0238 4 4 46 2 3 0144468999 10 2 039 3 1 67 2 ----+----+----+----+

22. 22  Useful for small data sets ◦ Less than 100 observations  Can also be used to compare groups ◦ Back-to-Back Stem and Leaf Plots, using the same stems for both groups.  Murder Rate Data from U.S. and Canada ◦ Note: it doesn’t really matter whether the smallest stem is at top or bottom of the table STA 291 Summer 2010 Lecture 322

23. 23STA 291 Summer 2010 Lecture 323 PRESIDENTAGEPRESIDENTAGEPRESIDENTAGE Washington67Fillmore74Roosevelt60 Adams90Pierce64Taft72 Jefferson83Buchanan77Wilson67 Madison85Lincoln56Harding57 Monroe73Johnson66Coolidge60 Adams80Grant63Hoover90 Jackson78Hayes70Roosevelt63 Van Buren79Garfield49Truman88 Harrison68Arthur56Eisenhower78 Tyler71Cleveland71Kennedy46 Polk53Harrison67Johnson64 Taylor65McKinley58Nixon81 Reagan 93 Ford 93 StemLeaf

24. 24  Discrete data ◦ Frequency distribution  Continuous data ◦ Grouped frequency distribution  Small data sets ◦ Stem and leaf plot  Interval data ◦ Histogram  Categorical data ◦ Bar chart ◦ Pie chart  Grouping intervals should be of same length, but may be dictated more by subject-matter considerations STA 291 Summer 2010 Lecture 324

25. 25  Present large data sets concisely and coherently  Can replace a thousand words and still be clearly understood and comprehended  Encourage the viewer to compare two or more variables  Do not replace substance by form  Do not distort what the data reveal STA 291 Summer 2010 Lecture 325

26. 26  Don’t have a scale on the axis  Have a misleading caption  Distort by using absolute values where relative/proportional values are more appropriate  Distort by stretching/shrinking the vertical or horizontal axis  Use bar charts with bars of unequal width STA 291 Summer 2010 Lecture 326

27. 27  Frequency distributions and histograms exist for the population as well as for the sample  Population distribution vs. sample distribution  As the sample size increases, the sample distribution looks more and more like the population distribution ◦ This will be explored further later on in the course STA 291 Summer 2010 Lecture 327

28. 28  The population distribution for a continuous variable is usually represented by a smooth curve ◦ Like a histogram that gets finer and finer  Similar to the idea of using smaller and smaller rectangles to calculate the area under a curve when learning how to integrate  Symmetric distributions ◦ Bell-shaped ◦ U-shaped ◦ Uniform  Not symmetric distributions: ◦ Left-skewed ◦ Right-skewed ◦ Skewed STA 291 Summer 2010 Lecture 328

29. Symmetric Right-skewed Left-skewed STA 291 Summer 2010 Lecture 329