Chapter 4 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 Q 3 median of upper half (upper half = 69 largest pulses); Q 3 = pulse in position 35 from the high end; Q 3 =78

Recall the 5-number summary of data from Chapter 4 n Minimum Q 1 median Q 3 maximum n Pulse data 5-number summary  A boxplot is a graphical display of the 5- number summary

Example n 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)

m = median = 3.4 Q 3 = third quartile = 4.2 Q 1 = first quartile = 2.3 Largest = max = 6.1 Smallest = min = 0.6 Five-number summary: min Q 1 m Q 3 max Boxplot: display of 5-number summary BOXPLOT

Boxplot: display of 5-number summary n Example: age of 66 “crush” victims at rock concerts number summary:

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

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

Q 3 = third quartile = 4.2 Q 1 = first quartile = 2.3 Largest = max = 7.9 Boxplot: display of 5-number summary BOXPLOT 8 Interquartile range Q 3 – Q 1 = 4.2 − 2.3 = 1.9 Distance to Q − 4.2 = * 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 Q 3. Thus, individual #25 is a suspected outlier.

ATM Withdrawals by Day, Month, Holidays

Beg. of class pulses (n=138) n Q 1 = 63, Q 3 = 78 n IQR=78  63=15 n 1.5(IQR)=1.5(15)=22.5 n Q (IQR): 63 – 22.5=40.5 n Q (IQR): =

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 Q 3 ? Pass Catching Yards by Receivers Countdown 10

Rock concert deaths: histogram and boxplot

Automating Boxplot Construction n Excel “out of the box” does not draw boxplots. n Many add-ins are available on the internet that give Excel the capability to draw box plots. n Statcrunch ( draws box plots.

Q 3 = third quartile = 4.2 Q 1 = first quartile = 2.3 Largest = max = 7.9 Statcrunch Boxplot

Tuition 4-yr Colleges

Statcrunch: NFL Salaries by Position

College Football Head Coach Salaries by Conference

2013 Major League Baseball Salaries by Team

End of Chapter 4