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Chapter 3: Variability
• Mean Sometimes Not Sufficient
• Frequency Distributions
• Normal Distribution
• Standard Deviation
What City has Temperatures to My Liking?
• Person 1: Likes Seasons and Variability
• Person 2: Likes Consistency, Cool Temps
Average
Temperature
by City
(1961-1990)
30
Duluth
Juneau
Bismarck
Burlington
Great Falls
Minneapolis-St. Paul
Portland
Sioux Falls
Spokane
Buffalo
Detroit
Chicago
Cleveland
Denver
Pittsburgh
Providence
Omaha
Boise
Boston
Salt Lake City
Seattle-Tacoma
Indianapolis
Kansas City
Portland
Philadelphia
New York 1
Baltimore
St. Louis
San Francisco
Washington
Nashville
Norfolk
Oklahoma City
Charlotte
Atlanta
Memphis
Los Angeles
Columbia
San Diego
Jackson
Dallas-Fort Worth
Houston
New Orleans
Phoenix
Miami
Honolulu
35
40
45
50
55
60
65
70
75
80
38.5
40.6
41.6
44.6
44.8
44.9
45.4
45.5
47.3
47.7
48.6
49
49.6
50.3
50.3
50.4
50.6
50.9
51.3
52
52
52.3
53.6
53.6
54.3
54.7
55.1
56.1
57.1
58
59.1
59.2
60
60.1
61.3
62.3
63
63.1
64.2
64.2
65.4
67.9
68.1
72.6
75.9
77.2
Temperature
Proximity to
Ocean
Elevation
Latitude:
South-North
Climate:
• Precipitation
• Humidity
Temperature Variation
Across Cities in 2011
Boston
30
60
90
San Francisco
30
60
90
San Diego
30
60
90
Austin
30
60
90
30
Tampa Bay
60
90
Similar Mean, Different Distributions
Seattle
Portland
Boston
Omaha
Normal Distribution
• Adolphe Quételet (1796-1874)
• ‘Quetelet Index’: Weight / Height
(“Body Mass Index”)
Normal Distribution
Two Metrics:
Mean and Standard Deviation
Calculating Standard Deviation
Mean
• A deviation is the difference between the mean and an actual data point.
• Deviations are calculated by taking each value and subtracting the mean:
deviation  xi  x
Summary the Deviation?
• Deviations cancel out because
some are positive and others
negative.
• Overall would be 0
• Not Useful
Sum of Squared Deviation
• Therefore, we square each deviation.
• We get the sum of squares (SS).
^2
Variance
• The sum of squares is a good measure of overall
variability, but is dependent on the number of scores
• We calculate the average variability by dividing by the
number of scores (n)
• This value is called the variance (s2)
Standard Deviation
• Variance is measured in units squared
• This isn’t a very meaningful metric so
we take the square root value.
• This is the standard deviation (s)
^2
Median
55
2
19
36
53
70
87
104
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