Download 3-1 and 3-2

Chapter 3
Statistics for Describing,
Exploring, and Comparing Data
3-1 Overview
3-2 Measures of Center
3-3 Measures of Variation
3-4 Measures of Relative Standing
3-5 Exploratory Data Analysis (EDA)
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Section 3-1
Overview
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Overview
 Descriptive Statistics
summarize or describe the important
characteristics of a known set of
data
 Inferential Statistics
use sample data to make inferences
(or generalizations) about a
population
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Section 3-2
Measures of Center
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Key Concept
When describing, exploring, and comparing
data sets, these characteristics are usually
extremely important: center, variation,
distribution, outliers, and changes over time.
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Definition
 Measure of Center
the value at the center or middle of a
data set
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Definition
Arithmetic Mean
(Mean)
the measure of center obtained by adding
the values and dividing the total by the
number of values
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Notation

denotes the sum of a set of values.
x
is the variable usually used to represent the
individual data values.
n
represents the number of values in a sample.
N
represents the number of values in a population.
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Notation
x is pronounced ‘x-bar’ and denotes the mean of a set
of sample values
x =
x
n
µ is pronounced ‘mu’ and denotes the mean of all values
in a population
µ =
x
N
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Definitions
 Median
the middle value when the original
data values are arranged in order of
increasing (or decreasing) magnitude
 often denoted by x~
(pronounced ‘x-tilde’)
 is not affected by an extreme value
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Finding the Median
 If the number of values is odd, the
median is the number located in the
exact middle of the list.
 If the number of values is even, the
median is found by computing the
mean of the two middle numbers.
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5.40
1.10
1.10
0.42
0.73
0.48
0.42
5.40
0.48
0.73
1.10
1.10
(in order - even number of values – no exact middle
shared by two numbers)
0.73 + 1.10
MEDIAN is 0.915
2
5.40
1.10
0.42
0.73
0.48
1.10
0.66
0.42
0.48
0.66
0.73
1.10
1.10
5.40
(in order - odd number of values)
exact middle
MEDIAN is 0.73
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Definitions
 Mode
the value that occurs most frequently
 Mode is not always unique
 A data set may be:
Bimodal
Multimodal
No Mode
Mode is the only measure of central tendency
that can be used with nominal data
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Mode - Examples
a. 5.40 1.10 0.42 0.73 0.48 1.10
Mode is 1.10
b. 27 27 27 55 55 55 88 88 99
Bimodal -
c. 1 2 3 6 7 8 9 10
No Mode
27 & 55
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Definition
 Midrange
the value midway between the maximum and
minimum values in the original data set
Midrange =
maximum value + minimum value
2
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Round-off Rule for
Measures of Center
Carry one more decimal place than is
present in the original set of values.
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You do!
Here are the volumes (in ounces) or randomly selected
cans of Coke:
12.3 12.1 12.2 12.3 12.2
Find the mean, median, mode, and midrange.
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Mean from a Frequency
Distribution
Assume that in each class, all sample
values are equal to the class
midpoint.
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Mean from a Frequency
Distribution
use class midpoint of classes for variable x
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Find the Mean from a Frequency
Distribution
Age of Actress
Frequency (f)
21-30
28
Class Midpoint f*x
(x)
25.5
714
31-40
30
35.5
1065
41-50
12
45.5
546
51-60
2
55.5
111
61-70
2
65.5
131
71-80
2
75.5
151
Totals
76
2718
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Weighted Mean
In some cases, values vary in their degree of
importance, so they are weighted accordingly.
 (w • x)
x =
w
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Example
• We have three test sores: 85, 90, 75
• The first test counts for 20%, the second for 30%,
and the third for 50% of the final grade.
• Find the weighted mean.
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Best Measure of Center
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Definitions
 Symmetric
distribution of data is symmetric if the
left half of its histogram is roughly a
mirror image of its right half
 Skewed
distribution of data is skewed if it is not
symmetric and if it extends more to
one side than the other
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Skewness
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Heart Rate Activity
Is there a difference between male and
female heart rates?
Male: 60 67 59 64 80 55 72 84 59 67 69
Female: 83 56 57 63 60 69 70 86 70 57 67 75
72 75 57 76 69 79 84 75 56
Compare: 1) The sample sizes
2) The Means and the Medians (centers)
3) The IQR, the 5 # Summaries and the
standard deviations (spreads)
4) Look at the shape: symmetric, skewed,
or irregular?
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