Copyright © 2009 Pearson Education, Inc. Slide 4- 1 Practice – Ch4 #26: A meteorologist preparing a talk about global warming compiled a list of weekly low temperatures (in degrees Fahrenheit) he observed at his south Florida home last year. The coldest temp. for any week was 36F, but he inadvertently recorded the Celsius value of 2 degrees. Assuming he correctly listed all the other temperatures, explain how this error will affect these summary statistics: Measures of center: mean and median Mean will be smaller, Median will not be affected Measures of spread: range, IQR, and standard deviation The range and standard deviation will be larger, the IQR won’t change.

Copyright © 2009 Pearson Education, Inc. Slide 1- 2 StatCrunch lab – ch 4 Open StatCrunch from the CourseCompass website Select s from the Chapter 4 data sets A university teacher saved every received from students in a large Introductory Statistics class during an entire term. She then counted, for each student who had sent her at least , how many s each student had sent. Create a histogram of the data – be sure to set bin-width at 1 and to start bins at 1 (why?) What are the appropriate labels for the X and Y axes? (remember, think first, then show, then tell) Given the histogram, would you expect the mean or median be larger? Why? Calculate summary statistics (Stat -> Summary Stats -> Columns) including the 5 number summary and mean and s.d. Describe the distribution in terms of shape (modes, symmetric/skewed, unusual features), center (median or mean), and spread (IQR or s.d.). Use complete sentences!

Copyright © 2009 Pearson Education, Inc. Chapter 5 Understanding and Comparing Distributions

Copyright © 2009 Pearson Education, Inc. Slide 3- 4 Objectives The student will be able to: Construct a boxplot by hand from a five-number summary. Describe the distribution of a quantitative variable with a description of the shape of the distribution, a numerical measure of center, and a numerical measure of spread. Compare the distributions of two or more groups by comparing their shapes, centers, and spreads. Compare two or more groups by comparing their boxplots. Use the 1.5 IQR rule to identify possible outliers.

Copyright © 2009 Pearson Education, Inc. Slide 1- 5 The Five-Number Summary Recall: the five-number summary of a distribution reports its median, quartiles, and extremes (maximum and minimum). Example: The five- number summary for the daily wind speed is: Max8.67 Q32.93 Median1.90 Q11.15 Min0.20

Copyright © 2009 Pearson Education, Inc. Slide 1- 6 Daily Wind Speed: Making Boxplots A boxplot is a graphical display of the five- number summary. Boxplots are particularly useful when comparing groups. Two kinds of boxplots – simple and modified. In a simple boxplot you only need to know the 5 number summary – whiskers will extend to the max and min. A modified boxplot extends whiskers to the max and min values within 1.5*IQR of the quartiles. Outliers are marked separately.

Copyright © 2009 Pearson Education, Inc. Practice Construct a simple and modified boxplot of our class data on # of siblings – Use your calculator to find the five number summary Then Draw the Boxplots by hand Slide 1- 7

Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Constructing Boxplots 1.Draw a single vertical axis spanning the range of the data. Draw short horizontal lines at the lower and upper quartiles and at the median. Then connect them with vertical lines to form a box.

Copyright © 2009 Pearson Education, Inc. Slide 1- 9 Constructing Boxplots (cont.) 2.Erect “fences” around the main part of the data. The upper fence is 1.5 IQRs above the upper quartile. The lower fence is 1.5 IQRs below the lower quartile. Note: the fences only help with constructing the boxplot and should not appear in the final display.

Copyright © 2009 Pearson Education, Inc. Slide Constructing Boxplots (cont.) 3.Use the fences to grow “whiskers.” Draw lines from the ends of the box up and down to the most extreme data values found within the fences. If a data value falls outside one of the fences, we do not connect it with a whisker.

Copyright © 2009 Pearson Education, Inc. Slide Constructing Boxplots (cont.) 4.Add the outliers by displaying any data values beyond the fences with special symbols. We often use a different symbol for “far outliers” that are farther than 3 IQRs from the quartiles.

Copyright © 2009 Pearson Education, Inc. Slide Wind Speed: Making Boxplots (cont.) Compare the histogram and boxplot for daily wind speeds: How does each display represent the distribution?

Copyright © 2009 Pearson Education, Inc. Slide Comparing Groups It is always more interesting to compare groups. With histograms, note the shapes, centers, and spreads of the two distributions. What does this graphical display tell you?

Copyright © 2009 Pearson Education, Inc. Slide Comparing Groups (cont) Boxplots offer an ideal balance of information and simplicity, hiding the details while displaying the overall summary information. We often plot them side by side for groups or categories we wish to compare. What do these boxplots tell you?

Copyright © 2009 Pearson Education, Inc. Slide What About Outliers? If there are any clear outliers and you are reporting the mean and standard deviation, report them with the outliers present and with the outliers removed. The differences may be quite revealing. Note: The median and IQR are not likely to be affected by the outliers.

Copyright © 2009 Pearson Education, Inc. Slide Examples Returning to our class data set – lets use our 5 number summary of number of siblings to construct a boxplot. Here are the weekly payrolls for two imaginary restaurants: Draw parallel boxplots (by hand) Write a few sentences comparing the distributions Where would you rather work and why? Moose- burgers McTofu

Copyright © 2009 Pearson Education, Inc. Slide Using the TI to make and compare boxplots Similar to plotting histograms Enter data in lists, To turn on stat plots, STATPLOT-> Plot1 -> ENTER Select On -> Enter select the first boxplot pictured (this is a modified boxplot and indicates outliers rather than a standard boxplot whose whiskers extend to the max and minimum), Xlist (L1 or L2) Frequency (will be 1 if all data is entered, may be another list if using a frequency table) Use Plot2 to display another data set Zoom -> 9 (ZoomStat) Use trace to explore the box plot Moose- burgers McTofu

Copyright © 2009 Pearson Education, Inc. Slide Using StatCrunch to make boxplots Enter StatCrunch from the textbook site Load data: Wines Graphs -> Boxplot Select Columns: CasePrice Group by: Location “Plot groups for each column” will create one graph with multiple boxplots “Plot columns for each group” will create multiple separate boxplots Click Next Check “Use fences to identify outliers” Fill in Y axis label Use the “Plot groups for each column” boxplot to answer these questions Which lake region produces the most expensive wine? Which lake region produces the lease expensive wine? In which region are the wines generally more expensive? Write a few sentences describing these wine prices?

Copyright © 2009 Pearson Education, Inc. Slide Practice Text #10, #25