2.4 Describing Distributions
Numerically
Numerical and More Graphical
Methods to Describe Univariate
Data
2 characteristics of a data set
to measure
center
measures where the “middle” of the
data is located
 variability
measures how “spread out” the data is

The median: a measure of
center
Given a set of n measurements arranged in
order of magnitude,
Median= middle value
n odd
mean of 2 middle values, n even
 Ex. 2, 4, 6, 8, 10; n=5; median=6
 Ex. 2, 4, 6, 8; n=4; median=(4+6)/2=5
Student Pulse Rates (n=62)
38, 59, 60, 60, 62, 62, 63, 63, 64, 64, 65, 67, 68, 70,
70, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75,
75, 75, 75, 76, 77, 77, 77, 77, 78, 78, 79, 79, 80, 80,
80, 84, 84, 85, 85, 87, 90, 90, 91, 92, 93, 94, 94, 95,
96, 96, 96, 98, 98, 103
Median = (75+76)/2 = 75.5
Medians are used often
Year 2016 baseball salaries
Median $1,956,250 (max=$32,000,000
Clayton Kershaw; min=$507,000)
 Median fan age: MLB 45; NFL 43; NBA
41; NHL 39
 Median existing home sales price: May
2011 $166,500; May 2010 $174,600
 Median household income (2008
dollars) 2009 $50,221; 2008 $52,029

The median splits the histogram
into 2 halves of equal area
Median Salaries by Major
Examples
Example: n = 7
17.5 2.8 3.2 13.9 14.1 25.3 45.8
 Example n = 7 (ordered): m = 14.1
 2.8 3.2 13.9 14.1 17.5 25.3 45.8
 Example: n = 8
17.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8
 Example n =8 (ordered) m = (14.1+17.5)/2 = 15.8
2.8 3.2 13.9 14.1 17.5 25.3 35.7 45.8

Below are the annual tuition charges at 7
public universities. What is the median
tuition?
4429
4960
4960
4971
5245
5546
7586
1.
2.
3.
4.
5245
4965.5
4960
4971
0%
1
0%
2.
0%
3
0%
4
10
Below are the annual tuition charges at 7
public universities. What is the median
tuition?
4429
4960
5245
5546
4971
5587
7586
1.
2.
3.
4.
5245
4965.5
5546
4971
0%
1
0%
2.
0%
3
0%
4
10
Measures of Spread

The range and interquartile
range
Ways to measure variability
range=largest-smallest
 OK sometimes; in general, too crude;
sensitive to one large or small data
value
 The range measures spread by
examining the ends of the data
 A better way to measure spread is to
examine the middle portion of the data
Quartiles: Measuring spread by
examining the middle
The first quartile, Q1, is the value in the
sample that has 25% of the data at or
below it (Q1 is the median of the lower
half of the sorted data).
The third quartile, Q3, is the value in the
sample that has 75% of the data at or
below it (Q3 is the median of the upper
half of the sorted data).
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
1
2
3
4
5
6
7
6
5
4
3
2
1
2
3
4
5
6
7
6
5
4
3
2
1
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
6.1
Q1= first quartile = 2.3
m = median = 3.4
Q3= third quartile = 4.2
Quartiles and median divide data
into 4 pieces
1/4
1/4
Q1
1/4
M
1/4
Q3
Quartiles are common
measures of spread

http://oirp.ncsu.edu/ir/admit

http://oirp.ncsu.edu/univ/peer

University of Southern California

Economic Value of College Majors
Mid-career
earnings by
major: 25th,
50th, 75th
percentiles.
Rules for Calculating Quartiles
Step 1: find the median of all the data (the median
divides the data in half)
Step 2a: find the median of the lower half; this median
is Q1;
Step 2b: find the median of the upper half; this
median is Q3.
Important:
when n is odd include the overall median in both
halves;
when n is even do not include the overall median in
either half.
11

Example
2 4 6 8 10 12 14 16 18 20
n = 10
Median
m
= (10+12)/2 = 22/2 = 11
Q1 :
Q3
median of lower half 2 4 6 8 10
Q1 = 6
: median of upper half 12 14 16 18 20
Q3 = 16
Pulse Rates n = 138
#
3
9
10
23
23
16
23
10
10
4
2
4
1
Stem
4*
4.
5*
5.
6*
6.
7*
7.
8*
8.
9*
9.
10*
10.
11*
Leaves
Median: mean of pulses in
locations 69 & 70:
median= (70+70)/2=70
588
001233444
5556788899
00011111122233333344444
55556666667777788888888
00000112222334444
55555666666777888888999
0000112224
5555667789
0012
58
0223
1
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
Below are the weights of 31 linemen on
the NCSU football team. What is the
value of the first quartile Q1?
1.
2.
3.
4.
287
257.5
263.5
262.5
#
stemleaf
2
2255
4
2357
6
2426
7
257
10
26257
12
2759
(4)
281567
15
2935599
10
30333
7
3145
5
32155
2
336
1
340
0%
1
0%
2.
0%
3.
0%
4.
Interquartile range
lower quartile Q1
 middle quartile: median
 upper quartile Q3
 interquartile range (IQR)
IQR = Q3 – Q1
measures spread of middle 50% of the
data

Example: beginning pulse
rates

Q3 = 78; Q1 = 63

IQR = 78 – 63 = 15
Below are the weights of 31 linemen on
the NCSU football team. The first quartile
Q1 is 263.5. What is the value of the IQR?
1.
2.
3.
4.
23.5
39.5
46
69.5
#
stemleaf
2
2255
4
2357
6
2426
7
257
10
26257
12
2759
(4)
281567
15
2935599
10
30333
7
3145
5
32155
2
336
1
340
0%
1.
0%
2.
0%
3
0%
4.
5-number summary of data

Minimum Q1 median Q3 maximum

Pulse data
45 63 70
78
111
Boxplot: display of 5-number summary
1
2
3
4
5
6
7
6
5
4
3
2
1
2
3
4
5
6
7
6
5
4
3
2
1
6.1
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
Largest = max = 6.1
BOXPLOT
7
Q3= third quartile
= 4.2
m = median = 3.4
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
4
3
2
1
Q1= first quartile
= 2.3
Smallest = min = 0.6
0
Disease X
Five-number summary:
min Q1 m Q3 max
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 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
Boxplot: display of 5-number summary
1
2
3
4
5
6
7
6
5
4
3
2
1
2
3
4
5
6
7
6
5
4
3
2
1
7.9
6.1
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
8
Largest = max = 7.9
7
BOXPLOT
Q3+1.5*IQR=
4.2+2.85 = 7.05
6
Q3= third quartile
= 4.2
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
Interquartile
range
Q3 – Q1=
4.2 − 2.3 =
1.9
4
3
2
1
0
Disease X
Q1= first quartile
= 2.3
1.5 * IQR = 1.5*1.9=2.85. Individual #25 has a value of
7.9 years, so 7.9 is an outlier. The line from the top
end of the box is drawn to the biggest number in the
data that is less than 7.05
ATM Withdrawals by Day,
Month, Holidays
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
63
45
70
78
100.5
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 ?
0
136
273
410
547
684
958
821
1095
1232
1369
Pass Catching Yards by Receivers
1.
2.
3.
4.
450
750
215
545
0%
1
0%
2
0%
3
0%
4
10
Rock concert deaths: histogram
and boxplot
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.

Statcrunch Boxplot
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
6
5
4
3
2
1
2
3
4
5
6
7
6
5
4
3
2
1
7.9
6.1
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
Largest = max = 7.9
Q3= third quartile
= 4.2
Q1= first quartile
= 2.3
Tuition 4-yr Colleges
Macro: Stock, bond returns-30 yrs
Smallest = -21.98
Q1 = 0.2075
Median = 2.935
Q3 = 10.725
Largest = 42.98
IQR = 10.5175
Outliers = (42.98,
-21.98)
Bonds BoxPlot
-40
Smallest = -26.61
Q1 = -0.555
10
Median = 10.43510
Q3 = 25.1275 10
Largest = 44.38 10
10
IQR = 25.6825 10
Outliers = () 10
-20
0
Stocks
x
44.38
34.84
32.54
29.93
29.22
27.31
25.30
10 -40.00
25.07
10
22.36
BoxPlot
10
10
10
10
10
10
10
-20.00
10
10
0.00
20
40
60
20.00
40.00
60.00
BoxPlot