1. Boxplots (Box and Whisker Plots)

2. Comparing Data Using Boxplots Each section of the boxplot represents 25% of the data. The median (50%tile) is the line in the middle of the box. The whiskers extend to the max and min value that aren’t outliers. Any outliers are dots past the end of the whiskers.

3. 5 Number Summary for a Boxplot –Minimum –Q1 (Quartile 1 – 25 th percentile) –Median (50 th percentile) –Q3 (Quartile 3 – 75 th percentile) –Maximum –These are the 5 main points on the boxplot

4. Finding the Median & Quartiles To find the median of a set of data:To find the median of a set of data: –Order the data from least to greatest –The median is the middle number –If there is an even number of numbers and there is no one middle number, then average the two middle numbers To find the Quartiles:To find the Quartiles: –Q1 is the median of the lower half of the data –Q3 is the median of the top half of the data –Never include the actual median in the data when finding the quartiles

5. How to Make a Boxplot Find the 5 Number SummaryFind the 5 Number Summary Scale the axis so all numbers fit appropriatelyScale the axis so all numbers fit appropriately Make the box start at Q1 and end at Q3Make the box start at Q1 and end at Q3 Draw a line in the box marking the medianDraw a line in the box marking the median Extend “whiskers” to the minimum and maximumExtend “whiskers” to the minimum and maximum –Modified Boxplot: If there are outliers, extend whiskers to the smallest and largest values that aren’t outliers and put dots where the outliers lie

6. Finding Outliers IQR or Interquartile Range = Q3 – Q1 An outlier on the low end is any point lower than Q1 - 1.5(IQR) An outlier on the high end is any point higher than Q3 + 1.5(IQR)

7. Make and compare Boxplots: Poverty Rates in the Eastern US Southern Poverty (%) Northern Maryland6.1 New Hampshire 4.3 Delaware6.5Wisconsin5.6 Florida9.0Connecticut6.2 North Carolina 9.0 New Jersey 6.3 Georgia9.9Vermont6.3 Tennessee10.3Indiana6.7 South Carolina 10.7Massachusetts6.7 Alabama12.5Michigan7.4 Kentucky12.7Maine7.8 Virginia13.9Ohio7.8 West Virginia 13.9Pennsylvania7.8 Mississippi16.0Illinois7.8 Rhode Island 8.9 New York 11.5

8. 5 Number Summary & Outliers Southern States Min: 6.1 Q1: 9.0 Median: 10.5 Q3: 13.3 Max: 16 Outliers: < 9.0 – 1.5(13.3-9.0) = 2.55 no poverty rates are < 2.55 > 13.3 + 1.5(13.3-9.0) = 19.75 no poverty rates are >19.75 so no outliers on either end Northern States Min: 4.3 Q1: 6.3 Median: 7.05 Q3: 7.8 Max: 11.5 Outliers: < 6.3 – 1.5(7.8-6.3) = 4.05 no poverty rates are < 4.05 > 7.8 + 1.5(7.8-6.3) = 10.05 NY is 11.5 which is >10.05 so NY is an outlier

10. Boxplots in Calculator Enter data into List (Stat Edit)Enter data into List (Stat Edit) Choose 1 st boxplot option in StatPlotChoose 1 st boxplot option in StatPlot Choose the list you used for XlistChoose the list you used for Xlist Choose 1 for Freq or a 2 nd list if data is stored in two lists (values in one, frequency in another)Choose 1 for Freq or a 2 nd list if data is stored in two lists (values in one, frequency in another) Zoom 9 will scale it for you to see the graphZoom 9 will scale it for you to see the graph Press Trace and the arrow keys to see the five number summary and any outliersPress Trace and the arrow keys to see the five number summary and any outliers

12. Measures of Center Mean( , ) —add up data values and divide by number of data values Median (M)—list data values in order, locate middle data value; average middle 2 if necessary Data Set: 19, 20, 20, 21, 22 Mean = 20.4; Median = 20 Data Set: 19, 20, 20, 21, 38 Mean = 23.6; Median = 20

13. Robust (Resistant) Statistic Robust or resistant: value doesn’t change dramatically when extreme values (including outliers) are added to (or taken out of) the data set.Robust or resistant: value doesn’t change dramatically when extreme values (including outliers) are added to (or taken out of) the data set. –Median is resistant. –Mean is NOT resistant against extreme values. Mean is pulled away from the center of the distribution toward the extreme value (“tails of graph”).

14. Mean or Median?

15. Measures of Center on Different Distribution Shapes Skewed to the left Symmetric Skewed to the right In each of the graphs, decide which mark represents the mean µ, median M, and mode Mo. Remember the mean is pulled toward extreme values. µ, M, Mo all = Mo, M, µ