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BPS - 5th Ed. Chapter 1 1
Chapter 1
Picturing Distributions with Graphs
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BPS - 5th Ed. Chapter 1 2
Statistics
Statistics is a science that involves the extraction of
information from numerical data obtained during an
experiment or from a sample. It involves the designof the experiment or sampling procedure, the
collection and analysis of the data, and making
inferences (statements) about the population based
upon information in a sample.
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BPS - 5th Ed. Chapter 1 3
Individuals and Variables
Individuals
the objects described by a set of data
may be people, animals, or thingsVariable
any characteristic of an individual
can take different values for differentindividuals
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BPS - 5th Ed. Chapter 1 4
Variables
Categorical
Places an individual into one of severalgroups or categories
Quantitative (Numerical)
Takes numerical values for whicharithmetic operations such as adding andaveraging make sense
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BPS - 5th Ed. Chapter 1 5
Case Study
The Effect of Hypnosison the
Immune System
reported in Science News, Sept. 4, 1993, p. 153
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BPS - 5th Ed. Chapter 1 6
Case Study
The Effect of Hypnosison the
Immune SystemObjective:
To determine if hypnosis strengthens the
disease-fighting capacity of immune cells.
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BPS - 5th Ed. Chapter 1 7
Case Study
65 college students.
33 easily hypnotized
32 not easily hypnotizedwhite blood cell counts measured
all students viewed a brief video about
the immune system.
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BPS - 5th Ed. Chapter 1 8
Case Study
Students randomly assigned to one ofthree conditions
subjects hypnotized, given mental exercise subjects relaxed in sensory deprivation
tank
control group (no treatment)
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BPS - 5th Ed. Chapter 1 9
Case Study
white blood cell counts re-measured after oneweek
the two white blood cell counts are compared
for each group
results
hypnotized group showed larger jump in white
blood cells easily hypnotized group showed largest immune
enhancement
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BPS - 5th Ed. Chapter 1 10
Case Study
Variables measured
Easyor difficultto achieve
hypnotic tranceGroup assignment
Pre-study white blood cell
count Post-study white blood cell
count
categorical
quantitative
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BPS - 5th Ed. Chapter 1 11
Case Study
Weight Gain Spells
Heart Risk for WomenWeight, weight change, and coronary heart disease
in women. W.C. Willett, et. al., vol. 273(6), Journalof the American Medical Association, Feb. 8, 1995.
(Reported in Science News, Feb. 4, 1995, p. 108)
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BPS - 5th Ed. Chapter 1 12
Case Study
Weight Gain Spells
Heart Risk for WomenObjective:
To recommend a range of body mass index
(a function of weight and height) in terms ofcoronary heart disease (CHD) risk in women.
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BPS - 5th Ed. Chapter 1 13
Case Study
Study started in 1976 with 115,818women aged 30 to 55 years and withouta history of previous CHD.
Each womans weight (body mass) was
determined.
Each woman was asked her weight atage 18.
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BPS - 5th Ed. Chapter 1 14
Case Study
The cohort of women were followed for14 years.
The number of CHD (fatal and nonfatal)cases were counted (1292 cases).
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BPS - 5th Ed. Chapter 1 15
Case Study
Age (in 1976)
Weight in 1976
Weight at age 18
Incidence of coronary heartdisease
Smoker or nonsmoker Family history of heart disease
quantitative
categorical
Variables measured
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BPS - 5th Ed. Chapter 1 16
Distribution
Tells what values a variable takes andhow often it takes these values
Can be a table, graph, or function
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BPS - 5th Ed. Chapter 1 17
Displaying Distributions
Categorical variables
Pie charts
Bar graphsQuantitative variables
Histograms
Stemplots (stem-and-leaf plots)
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BPS - 5th Ed. Chapter 1 18
Year Count Percent
Freshman 18 41.9%
Sophomore 10 23.3%
Junior 6 14.0%
Senior 9 20.9%
Total 43 100.1%
Data Table
Class Make-up on First Day
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BPS - 5th Ed. Chapter 1 19
Freshman41.9%
Sophomore
23.3%
Junior
14.0%
Senior
20.9%
Pie Chart
Class Make-up on First Day
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BPS - 5th Ed. Chapter 1 20
41.9%
23.3%
14.0%
20.9%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
45.0%
Freshman Sophomore Junior Senior
Year in School
Percent
Class Make-up on First Day
Bar Graph
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BPS - 5th Ed. Chapter 1 21
Example: U.S. Solid Waste (2000)
Data TableMaterial Weight (million tons) Percent of total
Food scraps 25.9 11.2 %
Glass 12.8 5.5 %
Metals 18.0 7.8 %
Paper, paperboard 86.7 37.4 %
Plastics 24.7 10.7 %
Rubber, leather, textiles 15.8 6.8 %
Wood 12.7 5.5 %
Yard trimmings 27.7 11.9 %
Other 7.5 3.2 %
Total 231.9 100.0 %
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BPS - 5th Ed. Chapter 1 22
Example: U.S. Solid Waste (2000)
Pie Chart
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BPS - 5th Ed. Chapter 1 23
Example: U.S. Solid Waste (2000)
Bar Graph
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BPS - 5th Ed. Chapter 1 24
Examining the Distribution of
Quantitative DataOverall pattern of graph
Deviations from overall pattern
Shape of the data
Center of the data
Spread of the data (Variation)Outliers
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BPS - 5th Ed. Chapter 1 25
Shape of the Data
Symmetric
bell shaped
other symmetric shapes
Asymmetric
right skewed
left skewed
Unimodal, bimodal
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BPS - 5th Ed. Chapter 1 26
Symmetric
Bell-Shaped
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BPS - 5th Ed. Chapter 1 27
Symmetric
Mound-Shaped
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BPS - 5th Ed. Chapter 1 28
Symmetric
Uniform
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BPS - 5th Ed. Chapter 1 29
Asymmetric
Skewed to the Left
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BPS - 5th Ed. Chapter 1 30
Asymmetric
Skewed to the Right
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BPS - 5th Ed. Chapter 1 31
Outliers
Extreme values that fall outside theoverall pattern
May occur naturally
May occur due to error in recording
May occur due to error in measuring
Observational unit may be fundamentally
different
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BPS - 5th Ed. Chapter 1 32
Histograms
For quantitative variables that takemany values
Divide the possible values into classintervals (we will only consider equal widths)
Count how many observations fall in
each interval(may change to percents)
Draw picture representing distribution
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BPS - 5th Ed. Chapter 1 33
Histograms: Class Intervals
How many intervals?
One rule is to calculate the square root of the
sample size, and round up.
Size of intervals?
Divide range of data (maxmin) by number of
intervals desired, and round to convenient number
Pick intervals so each observation can onlyfall in exactly one interval (no overlap)
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BPS - 5th Ed. Chapter 1 34
Case Study
Weight Data
Introductory Statistics classSpring, 1997
Virginia Commonwealth University
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BPS - 5th Ed. Chapter 1 35
Weight Data
192 110 195 180 170 215
152 120 170 130 130 125
135 185 120 155 101 194
110 165 185 220 180
128 212 175 140 187180 119 203 157 148
260 165 185 150 106
170 210 123 172 180
165 186 139 175 127
150 100 106 133 124
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BPS - 5th Ed. Chapter 1 36
Weight Data: Frequency Table
Weight Group Count100 -
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BPS - 5th Ed. Chapter 1 37
Weight Data: Histogram
0
2
4
6
8
10
12
14
Frequency
100 120 140 160 180 200 220 240 260 280Weight
* Left endpoint is included in the group, right endpoint is not.
Numberofstudents
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BPS - 5th Ed. Chapter 1 38
Stemplots(Stem-and-Leaf Plots)
For quantitative variables
Separate each observation into a stem (firstpart of the number) and a leaf (the remaining
part of the number)
Write the stems in a vertical column; draw avertical line to the right of the stems
Write each leaf in the row to the right of itsstem; order leaves if desired
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BPS - 5th Ed. Chapter 1 39
Weight Data
192 110 195 180 170 215
152 120 170 130 130 125
135 185 120 155 101 194
110 165 185 220 180
128 212 175 140 187180 119 203 157 148
260 165 185 150 106
170 210 123 172 180
165 186 139 175 127
150 100 106 133 124
1
2
10
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BPS - 5th Ed. Chapter 1 40
Weight Data:
Stemplot(Stem & Leaf Plot)
10
11
12
13
1415
16
17
18
19
20
21
22
23
24
25
26
Key
20|3 means
203 pounds
Stems = 10sLeaves = 1s
192
2
1522
5
135
10 0166
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BPS - 5th Ed. Chapter 1 41
Weight Data:
Stemplot(Stem & Leaf Plot)
10 0166
11 009
12 0034578
13 00359
14 0815 00257
16 555
17 000255
18 000055567
19 245
20 3
21 025
22 0
23
24
25
26 0
Key
20|3 means
203 pounds
Stems = 10sLeaves = 1s
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BPS - 5th Ed. Chapter 1 42
Extended Stem-and-Leaf Plots
If there are very few stems (when the
data cover only a very small range of
values), then we may want to create
more stems by splitting the original
stems.
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BPS - 5th Ed. Chapter 1 43
Extended Stem-and-Leaf Plots
Example: if all of the data values were
between 150 and 179, then we may
choose to use the following stems:151516
161717
Leaves 0-4 would go on eachupper stem (first 15), and leaves
5-9 would go on each lower stem(second 15).
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BPS - 5th Ed. Chapter 1 44
Time Plots
A time plot shows behavior over time. Time is always on the horizontal axis, and the
variable being measured is on the vertical axis.
Look for an overall pattern (trend), anddeviations from this trend. Connecting the data
points by lines may emphasize this trend.
Look for patterns that repeat at known regularintervals (seasonal variations).
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BPS - 5th Ed. Chapter 1 45
Class Make-up on First Day(Fall Semesters: 1985-1993)
0%
10%
20%
30%
40%
50%
60%
70%
Percent of Class
That Are Freshman
1985 1986 1987 1988 1989 1990 1991 1992 1993
Year of Fall Semester
Class Make-up On First Day
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Average Tuition (Public vs. Private)