Chapter 1
Picturing Distributions with Graphs
BPS - 5th Ed.
Chapter 1
1
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 design
of 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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Individuals and Variables
 Individuals
– the objects described by a set of data
– may be people, animals, or things
 Variable
– any characteristic of an individual
– can take different values for different
individuals
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Variables
 Categorical
– Places an individual into one of several
groups or categories
 Quantitative
(Numerical)
– Takes numerical values for which
arithmetic operations such as adding and
averaging make sense
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Distribution
 Tells
what values a variable takes and
how often it takes these values
 Can
be a table, graph, or function
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Displaying Distributions
 Categorical
variables
– Pie charts
– Bar graphs
 Quantitative
variables
– Histograms
– Stemplots (stem-and-leaf plots)
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Histograms
 For
quantitative variables that take
many values
 Divide the possible values into class
intervals (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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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 (maxmin) by number of
intervals desired, and round to convenient number

Pick intervals so each observation can only
fall in exactly one interval (no overlap)
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Stemplots
(Stem-and-Leaf Plots)
 For
quantitative variables
 Separate each observation into a stem (first
part of the number) and a leaf (the remaining
part of the number)
 Write the stems in a vertical column; draw a
vertical line to the right of the stems
 Write each leaf in the row to the right of its
stem; order leaves if desired
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1
2
192
152
135
110
128
180
260
170
165
150
BPS - 5th Ed.
Weight Data
110
120
185
165
212
119
165
210
186
100
195
170
120
185
175
203
185
123
139
106
Chapter 1
180
130
155
220
140
157
150
172
175
133
170
130
101
180
187
148
106
180
127
124
215
125
194
10
Weight Data:
Stemplot
(Stem & Leaf Plot)
Key
20|3 means
203 pounds
Stems = 10’s
Leaves = 1’s
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11
12
13 5
14
15 2
16
17
18
19 2
20
21
22
23
24
25
26
192
152
135
11
Weight Data:
Stemplot
(Stem & Leaf Plot)
Key
20|3 means
203 pounds
Stems = 10’s
Leaves = 1’s
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Chapter 1
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11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
0166
009
0034578
00359
08
00257
555
000255
000055567
245
3
025
0
0
12
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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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:
15
15
16
16
17
17
BPS - 5th Ed.
Leaves 0-4 would go on each
upper stem (first “15”), and leaves
5-9 would go on each lower stem
(second “15”).
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Examining the Distribution of
Quantitative Data
 Overall
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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Shape of the Data
 Symmetric
– bell shaped
– other symmetric shapes
 Asymmetric
– right skewed
– left skewed
 Unimodal,
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bimodal
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Symmetric
Bell-Shaped
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Symmetric
Mound-Shaped
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Symmetric
Uniform
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Asymmetric
Skewed to the Left
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Asymmetric
Skewed to the Right
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Outliers
 Extreme
values that fall outside the
overall 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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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), and
deviations from this trend. Connecting the data
points by lines may emphasize this trend.

Look for patterns that repeat at known regular
intervals (seasonal variations).
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Average Tuition (Public vs. Private)
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