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Chapter 12
Describing Distributions with
Numbers
Chapter 12
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Thought Question 1
Suppose you are helping to prepare a
meal for the homeless at a local shelter,
and you are told that on average about
25 people attend such meals. Is there
anything else you should know about the
data, in addition to the average, before
deciding how much food to prepare?
(Hint: Do 25 people attend every meal?)
Chapter 12
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Thought Question 2
The 2000-2001 Los Angeles Lakers
professional basketball team had a
median salary of $1,760,000 and an
average salary of $3,459,000. How are
these values computed? Which do you
think is a better measure of the typical
salary of a player on this team, the
median or the average?
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Thought Question 3
The length of human pregnancies has
a mean, or average, of 266 days for the
entire population of women. It also has
a standard deviation of 16 days.
What do you think is meant by the term
“standard deviation”?
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Turning Data Into Information
 Center
of the data
– mean
– median
– mode
 Spread
of the data (Variability)
– variance
– standard deviation
– range
– interquartile range
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Average or Mean ( X )
 Traditional
measure of center
 Sum the values and divide by the
number of values
n
1
1
x   x1  x 2  xn    xi
n
n i 1
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Median (M)
 A resistant
measure of the data’s center
 At least half of the ordered values are
less than or equal to the median value
 At least half of the ordered values are
greater than or equal to the median value
If n is odd, the median is the middle ordered value
 If n is even, the median is the average of the two
middle ordered values

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Median

Example 1 data: 2 4 6
Median (M) = 4

Example 2 data: 2 4 6 8
Median = 5 (ave. of 4 and 6)

Example 3 data: 6 2 4
Median
2
(order the values: 2 4 6 , so Median = 4)
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10
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Measures of Center
14
mean 15
16
median 17
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STAT 208 Class Survey
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Spring, 1997
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Virginia Commonwealth University
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Weight Data:
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Comparing the Mean & Median
 The
mean and median of data from a
symmetric distribution should be close
together. The actual (true) mean and
median of a symmetric distribution are
exactly the same.
 In a skewed distribution, the mean is
farther out in the long tail than is the
median [the mean is ‘pulled’ in the
direction of the possible outlier(s)].
Chapter 12
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Case Study
Airline fares
appeared in the New York Times on November 5, 1995
“...about 60% of airline passengers ‘pay less
than the average fare’ for their specific flight.”

How can this be?
13% of passengers pay more than 1.5 times
the average fare for their flight
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Spread or Variability
 If
all values are the same, then they all
equal the mean. There is no spread.
 Variability
exists when some values are
different from (above or below) the mean.
 We
will discuss the following measures of
spread: variance, standard deviation,
range, and interquartile range
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Variance and Standard Deviation
 When
variability exists, each data value
has an associated deviation from the
mean: xi  x
 What is a typical deviation from the
mean? (standard deviation)
 Small values of this typical deviation
indicate small spread in the data
 Large values of this typical deviation
indicate large spread in the data
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Variance
 Find
the mean
 Find the deviation of each value from
the mean
 Square the deviations
 Sum the squared deviations
 Divide the sum by n-1
(gives typical squared deviation from mean)
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Variance Formula
n
1
2
2
s 
( xi  x )

(n  1) i 1
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Standard Deviation Formula
typical deviation from the mean
n
1
2
s
( xi  x )

(n  1) i 1
[ standard deviation = square root of the variance ]
Click for
Computation
Chapter 12
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Traditional Summary Statistics
Weight Data
 Mean
= 158.75
 Standard deviation = 35.65
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Quartiles
 Three
numbers that divide the ordered
data into four equal sized groups.
 Q1
has 25% of the data below it.
 Q2 has 50% of the data below it. (Median)
 Q3 has 75% of the data below it.
Chapter 12
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Quartiles
Uniform Histogram
1st Qtr
Q1 2nd Qtr Q2
3rd Qtr
Chapter 12
Q3
4th Qtr
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Obtaining the Quartiles
 Order
the data.
 For Q2, just find the median.
 For Q1, look at the lower half of the data
values, those to the left of the median;
find the median of this lower half.
 For Q3, look at the upper half of the data
values, those to the right of the median;
find the median of this upper half.
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Weight Data: Sorted
100
101
106
106
110
110
119
120
120
123
124
125
127
128
130
130
133
135
139
140
148
150
150
152
155
157
165
165
165
170
170
170
172
175
175
180
180
180
180
185
Chapter 12
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185
186
187
192
194
195
203
210
212
215
220
260
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Weight Data: Quartiles
 Q 1=
127.5
 Q2= 165 (Median)
 Q3= 185
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10
11
12
first quartile 13
Quartiles
14
15
16
median or second quartile
17
third quartile 18
19
20
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22
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Weight Data:
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Five-Number Summary
 minimum
= 100
 Q1 = 127.5
 M = 165
 Q3 = 185
 maximum = 260
Interquartile
Range (IQR)
= Q3  Q1
= 57.5
IQR gives spread of middle 50% of the data
( “middle 50% of data ranges from Q1 to Q3” )
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Range
 One
way to measure spread is to give
the smallest (minimum) and largest
(maximum) values in the data set;
Range = max  min
( “the values range from min to max” )
 The
range is strongly affected by outliers,
and is rarely used
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Boxplot
(from Five-Number Summary)
 Central
 A line
box spans Q1 and Q3.
in the box marks the median M.
 Lines
extend from the box out to the
minimum and maximum.
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Weight Data: Boxplot
min
100
Q1
125
M
150
Q3
175
max
200
225
250
275
Weight
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Choosing a Summary
 Outliers
affect the values of the mean and
standard deviation.
 The five-number summary should be used to
describe center and spread for skewed
distributions, or when outliers are present.
 Use the mean and standard deviation for
reasonably symmetric distributions that are
free of outliers.
Chapter 12
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Number of Books Read for
Pleasure: Sorted
0
0
0
0
0
0
0
0
0
1
1
1
1
1
2
2
2
2
2
2
2
2
2
3
3
3
Med
3
4
4
4
4
4
4
5
5
5
5
5
5
6
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12
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14
14
15
15
20
20
30
99
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Five-Number Summary, Boxplot
Median = 3
interquartile range (iqr) = 5.5-1.0 = 4.5
range = 99-0 = 99
0
10
20
30
40
50
60
Number of books
Mean = 7.06
70
80
90
100
s.d. = 14.43
Chapter 12
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Number of Books Read for
Pleasure
10
9
8
7
6
5
4
3
2
1
0
00
7
14
21
28
35
42 49 56 63
Number of Books
Chapter 12
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77
84
91
98
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Key Concepts
 Numerical
Summaries
– Center (mean, median)
– Spread (variance, std. dev., range, IQR)
– Five-number summary & Boxplots
 Choosing
mean versus median
 Choosing standard deviation versus
five-number summary
Chapter 12
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Variance and Standard Deviation
Example
The following data consist of the metabolic rates
(cal./24hr.) of 7 men from a dieting study:
1792
1666
1362
1614
1460
1867
1439
First, compute the sample mean:
1792  1666  1362  1614  1460  1867  1439
x
7
11,200

7
 1600
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Variance and Standard Deviation
Example
Observations
Deviations
Squared deviations
xi  x 
xi
xi  x
1792
17921600 = 192
1666
1666 1600 =
1362
1362 1600 = -238
1614
1614 1600 =
1460
1460 1600 = -140
(-140)2 = 19,600
1867
1867 1600 = 267
(267)2 = 71,289
1439
1439 1600 = -161
(-161)2 = 25,921
sum =
2
66
14
0
Chapter 12
(192)2 = 36,864
(66)2 =
4,356
(-238)2 = 56,644
(14)2 =
196
sum = 214,870
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Variance and Standard Deviation
Example
214,870
s 
 35,811.67
7 1
2
s  35,811.67  189.24 calories
Return to
Slide 16
Chapter 12
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