Pulse Rates n = 138
# Stem Leaves4*
3 4. 5889 5* 00123344410 5. 555678889923 6* 0001111112223333334444423 6. 5555666666777778888888816 7* 0000011222233444423 7. 5555566666677788888899910 8* 000011222410 8. 55556677894 9* 00122 9. 584 10* 0223
10.1 11* 1
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
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
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)
25 1 6.124 2 5.623 3 5.322 4 4.921 5 4.720 6 4.519 7 4.218 6 4.117 5 3.916 4 3.815 3 3.714 2 3.613 1 3.412 2 3.311 3 2.910 4 2.89 5 2.58 6 2.37 7 2.36 6 2.15 5 1.54 4 1.93 3 1.62 2 1.21 1 0.6
m = median = 3.4
Q3= third quartile = 4.2
Q1= first quartile = 2.3
25 1 6.124 2 5.623 3 5.322 4 4.921 5 4.720 6 4.519 7 4.218 6 4.117 5 3.916 4 3.815 3 3.714 2 3.613 1 3.412 2 3.311 3 2.910 4 2.89 5 2.58 6 2.37 7 2.36 6 2.15 5 1.54 4 1.93 3 1.62 2 1.21 1 0.6
Largest = max = 6.1
Smallest = min = 0.6
Disease X
0
1
2
3
4
5
6
7
Yea
rs u
nti
l dea
th
Five-number summary:
min Q1 m Q3 max
Boxplot: display of 5-number summary
BOXPLOT
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
Boxplot construction1) 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
Q3= third quartile = 4.2
Q1= first quartile = 2.3
25 1 7.924 2 6.123 3 5.322 4 4.921 5 4.720 6 4.519 7 4.218 6 4.117 5 3.916 4 3.815 3 3.714 2 3.613 1 3.412 2 3.311 3 2.910 4 2.89 5 2.58 6 2.37 7 2.36 6 2.15 5 1.54 4 1.93 3 1.62 2 1.21 1 0.6
Largest = max = 7.9
Boxplot: display of 5-number summary
BOXPLOT
Disease X
0
1
2
3
4
5
6
7
Yea
rs u
nti
l dea
th
8
Interquartile range
Q3 – Q1=4.2 − 2.3 =
1.9
Distance to Q3
7.9 − 4.2 = 3.7
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.
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
7063 7840.5 100.545
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 ?
1 2 3 4
0% 0%0%0%
0 136273
410547
684821
9581095
12321369
Pass Catching Yards by Receivers
1. 450
2. 750
3. 215
4. 545CountdownCountdown
10
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.
Q3= third quartile = 4.2
Q1= first quartile = 2.3
25 1 7.924 2 6.123 3 5.322 4 4.921 5 4.720 6 4.519 7 4.218 6 4.117 5 3.916 4 3.815 3 3.714 2 3.613 1 3.412 2 3.311 3 2.910 4 2.89 5 2.58 6 2.37 7 2.36 6 2.15 5 1.54 4 1.93 3 1.62 2 1.21 1 0.6
Largest = max = 7.9
Statcrunch Boxplot
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