14.5 Box-and-Whisker Plots

1. 14.5 Box-and-Whisker Plots CORD Math Mrs. Spitz Spring 2007

2. Lesson Objectives: • Display and interpret data on a box-and-whisker plot. Assignment • pp. 581-582 #1-21 all

3. Data can be displayed in many ways. One method of displaying a set of data is with a box-and-whisker plot. Box-and-whisker plots are helpful in interpreting the distribution of data.

4. Let’s review a bit, shall we • We know that the median of a set of data separates the data into two equal parts.   Data can be further separated into quartiles. • The first quartile is the median of the lower part of the data.

5. Let’s review a bit, shall we • The second quartile is another name for the median of the entire set of data.The third quartile is the median of the upper part of the data. • Quartiles separate the original set of data into four equal parts.  Each of these parts contains one-fourth of the data.

6. Constructing a box-and-whisker plot:

7. Constructing a box-and-whisker plot:

8. Constructing a box-and-whisker plot:

9. Special Case: You may see a box-and-whisker plot, like the one below, which contains an asterisk. Sometimes there is ONE piece of data that falls well outside the range of the other values.  This single piece of data is called an outlier.  If the outlier is included in the whisker, readers may think that there are grades dispersed throughout the whole range from the first quartile to the outlier. 

10. Note: • Box-and-whisker plots are sometimes called box plots. • When analyzing a set of data, it is often helpful to draw a graphical representation of the data. One such graph which shows the quartiles and extreme values of the data is called a box-and-whisker plot.

11. Remember Mr. Juarez? • Mr. Juarez was the teacher in 14.2 whose scores we used to discuss some of the topics. He had 35 students in his 1st period Biology class. • Suppose Mr. Juarez wants to display this data in a box-and-whisker plot. First, he must arrange the data in numerical order or make a stem-and-leaf plot. The stem-and-leaf plot that we made is shown on the next slide.

12. Here’s the data set • First order the numbers • Make a stem-and-leaf plot • Find the median, lower quartile, upper quartile, and least value (LV) and greatest value (GV).

13. Stem-and-Leaf Plot • There are 35 values, so the median number will be 35/2 or 17.5 or the 18th number. That value is 80. 17/2 is 8.5 or 9 from the bottom. 8|2 represents a score of 82

14. Stem-and-Leaf Plot • 17/2 is 8.5 or 9 from the bottom will determine your lower quartile which is 65. • Count 9 from the top and that will determine your upper quartile which is 89. 8|2 represents a score of 82

15. Extreme Values • To determine your extreme values; • The least value (LV) is simply the lowest value. The lowest value in this case is 44. • To determine your greatest value (GV), look to see what is the greatest value. The greatest value in this case is 99.

16. Construct the box-and-whisker plot • First, draw a number line. Assign a scale to the number line that includes the extreme values. Plot dots to represent the extreme values (LV) and (GV), the upper and lower quartiles (LQ) and (UQ), and the median (M). LV UQ LQ M GV 40 50 60 70 90 80 100

17. Construct the box-and-whisker plot • Next, draw a box around the interquartile range. Mark the median by a vertical line through its point in the box. The median line will not always divide the box into equal parts. Draw a segment from the lower quartile to the greatest value. These segments are the whiskers of the plot. 40 50 60 70 90 80 100

18. About the whiskers . . . • Even though the whiskers are different lengths, each whisker contains at least one fourth of the data while the box contains at least one half of the data. • Now check for outliers. The interquartile range (IQ) is the lower quartile (LQ) subtracted from the upper quartile (UQ). 89 – 65 = 24

19. Outliers • Now check for outliers. The interquartile range (IQ) is the lower quartile (LQ) subtracted from the upper quartile (UQ). UQ – LQ = 89 – 65 = 24 To find the outliers, you multiply the IQ by 1.5 IQ ● 1.5 = 24 ● 1.5 = 36

20. 29 125 Outliers Now you need to take this value and subtract it from the LQ and add it to the UQ. Remember: LQ = 65 and UQ = 89 65 – 36 = 29 and 89 + 36 = 125 Any scores less than 29 or greater than 125 will be considered outliers. We don’t have any outliers in this case

21. But what if there are? • If a set of data contains outliers, then a box-and-whisker plot can be altered to show them. This is shown in the following example.

22. Example 2 • Twelve members of the Beck High School Pep Club are selling programs of the football game. The number of programs sold by each person is listed below:

23. Example 2 • Make a box-and-whisker plot of this data. • First arrange the data in numerical order. {23, 27, 39, 46, 46, 51, 53, 54, 55, 60, 69, 81} • Find the extreme values (LV) and (GV), so that would be 23 and 81.

24. Ex. 2 continued • Find the median. In this case because it is an even number of values, take the two middle values of 51 + 53/2 = 52. • Now find the lower quartile (LQ) which is 39 + 46/2 or 42.5. • The upper quartile is 55 + 60/2 or 57.5. Thus the IQ range is 57.5 – 42.5 or 15.

25. 20 80 Ex. 2 continued • Now check for outliers 57.5 + (1.5)(15) = 57.5 + 22.5 = 80 42.5 - (1.5)(15) = 42.5 - 22.5 = 20 That means: Any scores less than 20 or greater than 80 will be considered outliers. So, we don’t have anything less than 20, but we do have one value greater than 80 which is 81 which is an outlier.

26. Now let’s draw it 20 30 Median (M) = 52 Extreme Values LV = 23 and GV = 80 Lower Quartile (LQ) = 42.5 Upper Quartile (UQ) = 57.5 Outlier = 81 A point is plotted for 69 since it is the last value THAT IS NOT AN OUTLIER. The whisker is drawn to this point as shown. Outliers are plotted as isolated points. 40 50 60 70 80

27. b. Analyze the box-and-whisker plot to determine if any of the members did an exceptional job selling programs. • Based on this plot, Danny did an exceptionally good job selling programs.