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Chapter 5 Understanding and Comparing Distributions

Chapter 5 Understanding and Comparing Distributions. Another Useful Graphical Method: Boxplots. Pulse Rates n = 138. Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70. Q 1 : median of lower half (lower half = 69 smallest pulses); Q 1 = pulse in ordered position 35; Q 1 = 63.

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Chapter 5 Understanding and Comparing Distributions

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  1. Chapter 5 Understanding and Comparing Distributions Another Useful Graphical Method: Boxplots

  2. Pulse Rates n = 138 Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70 Q1: median of lower half (lower half = 69 smallest pulses); Q1 = pulse in ordered position 35; Q1 = 63 Q3 median of upper half (upper half = 69 largest pulses); Q3= pulse in position 35 from the high end; Q3=78

  3. Recall the 5-number summary of data from Chapter 4 • Minimum Q1 median Q3 maximum • Pulse data 5-number summary 45 63 70 78 111 • A boxplot is a graphical display of the 5-number summary

  4. Example • Consider the data shown at the left. • The data values 6.1, 5.6, …, are in the right column • They are arranged in decreasing order from 6.1 (data rank of 25 shown in far left column) to 0.6 (data rank of 1 in far left column) • The center column shows the ranks of the quartiles (in blue) from each end of the data and from the overall median (in yellow)

  5. Boxplot: display of 5-number summary Largest = max = 6.1 BOXPLOT Q3= third quartile = 4.2 m = median = 3.4 Q1= first quartile = 2.3 Five-number summary: min Q1 m Q3 max Smallest = min = 0.6

  6. Boxplot: display of 5-number summary • Example: age of 66 “crush” victims at rock concerts 1999-2000. 5-number summary: 13 17 19 22 47

  7. Boxplot construction 1) construct box with ends located at Q1 and Q3; in the box mark the location of median (usually with a line or a “+”) 2) fences are determined by moving a distance 1.5(IQR) from each end of the box; 2a) upper fence is 1.5*IQR above the upper quartile 2b) lower fence is 1.5*IQR below the lower quartile Note: the fences only help with constructing the boxplot; they do not appear in the final boxplot display

  8. Box plot construction (cont.) 3) whiskers: draw lines from the ends of the box left and right to the most extreme data values found within the fences; 4) outliers: special symbols represent each data value beyond the fences; 4a) sometimes a different symbol is used for “far outliers” that are more than 3 IQRs from the quartiles

  9. 8 Boxplot: display of 5-number summary Largest = max = 7.9 BOXPLOT Distance to Q3 7.9 − 4.2 = 3.7 Q3= third quartile = 4.2 Interquartile range Q3 – Q1= 4.2 − 2.3 = 1.9 Q1= first quartile = 2.3 1.5 * IQR = 1.5*1.9=2.85. Individual #25 has a value of 7.9 years, which is 3.7 years above the third quartile. This is more than 2.85 = 1.5*IQR above Q3. Thus, individual #25 is a suspected outlier.

  10. ATM Withdrawals by Day, Month, Holidays

  11. Beg. of class pulses (n=138) • Q1 = 63, Q3 = 78 • IQR=78  63=15 • 1.5(IQR)=1.5(15)=22.5 • Q1 - 1.5(IQR): 63 – 22.5=40.5 • Q3 + 1.5(IQR): 78 + 22.5=100.5 40.5 70 78 100.5 63 45

  12. Below is a box plot of the yards gained in a recent season by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q3 ? 410 958 136 684 1232 0 273 1369 821 547 1095 Pass Catching Yards by Receivers • 450 • 750 • 215 • 545 10 Countdown

  13. Rock concert deaths: histogram and boxplot

  14. Automating Boxplot Construction • Excel “out of the box” does not draw boxplots. • Many add-ins are available on the internet that give Excel the capability to draw box plots. • Statcrunch (http://statcrunch.stat.ncsu.edu) draws box plots.

  15. Statcrunch Boxplot Largest = max = 7.9 Q3= third quartile = 4.2 Q1= first quartile = 2.3

  16. Tuition 4-yr Colleges

  17. Statcrunch: 2012-13 NFL Salaries by Position

  18. College Football Head Coach Salaries by Conference

  19. 2013 Major League Baseball Salaries by Team

  20. End of Chapter 5

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