1.2 Displaying Quantitative Data w/ Graphs
Pages 25-48
Objectives
SWBAT:
1) Make and interpret dotplots and stemplots of
quantitative data.
2) Describe the overall pattern (shape, center, and
spread) of a distribution and identify any major
departures from the pattern (outliers).
3) Identify the shape of a distribution from a graph as
roughly symmetric or skewed.
4) Make and interpret histograms of quantitative data.
5) Compare distributions of quantitative data using
dotplots, stemplots, and histograms.
4) What is the most important
difference between cities C, F, and
G?
Their shapes
The dotplots show the daily
high temperatures for 7 cities
in June, July and August.
1) What is the most important
difference between cities A, B,
and C?
Their centers
2) What is the most important
difference between cities C and
D?
Their spreads
3) What are two important
differences between cities D
and E?
Their spreads (but not range)
and unusual values (outliers)
When describing the distribution of a quantitative
variable, what characteristics should be addressed?
• You want to address patterns and departures
from patterns. The acronym to remember is
SOCS: Shape, Outliers, Center, and Spread.
Someone else
is on 1.2!
Shape
• When you describe a distribution’s shape, concentrate on the
main features. Look for rough symmetry or clear skewness.
Definitions:
A distribution is roughly symmetric if the right and left sides of the
graph are approximately mirror images of each other.
A distribution is skewed to the right (right-skewed) if the right side of
the graph (containing the half of the observations with larger values) is
much longer than the left side.
It is skewed to the left (left-skewed) if the left side of the graph is
much longer than the right side.
Symmetric
Skewed-left
Skewed-right
To help remember skewed right and skewed left, think
about your feet.
A distribution is skewed to the right
when the right side of the graph is
more spread out than the left side.
Think about your right foot. The
toes are tall on the left side and get
progressively smaller as you move
to the right.
A distribution is skewed to the left
when the left side of the graph is
more spread out than the right side.
Think about your left foot. The toes
are tall on the right side and get
progressively smaller as you move
to the left.
Other terms to describe shape
Unimodal
A distribution is unimodal when it shows one
distinct peak.
Bimodal
A distribution is bimodal if it has two distinct peaks. Note: we
don’t worry about little bumps. They have to be distinct.
This is an example of
a bimodal
distribution. It shows
the duration (in
minutes) of 220
eruptions of the Old
Faithful geyser.
Uniform
A distribution is uniform when the heights of the
bars are all about the same.
How would you describe the shapes of these distributions?
Skewed right,
unimodal
Symmetric,
unimodal
• An outlier is an individual value that falls
outside the overall pattern of a distribution.
– For now we’ll use an eye test to determine
outliers.
Looking at this distribution, there’s two unusually
high values that appear to be outliers, at
approximately 57 and 91.
• The center is the middle value in the
distribution (either the mean or median).
• The spread is the variability of a sampling
distribution (how spread out the data is).
– Common measures of spread are range and IQR.
• Here is an example of Tom Brady’s passer
ratings in the 2001 NFL season.
Describe the spread.
The range is 148.3-57.1=91.2
Frozen Pizza Example
Here are the number of calories per serving for 16
brands of frozen cheese pizza, along with a dotplot
of the data.
340 340 310 320 310 360 350 330
260 380 340 320 360 290 320 330
Shape: roughly symmetric and unimodal
Center: median at 330 calories
Spread: the values vary from 260 calories to 380 calories (a range of
120)
Outliers: there appears to be one unusually small value (260 calories)
What is the most important thing to remember when
you are asked to compare two distributions?
• You need to actually compare the distributions
using explicit comparison words!
– Examples:
• The center for distribution A is larger than the center for
distribution B.
• Carucci’s cat meows less than Mr. Fal’s cat.
• Prestige Worldwide makes the same amount of money
as the South Pole Elf Corporation.
• Needless to say, this is only applicable to certain
parts of SOCS (center and spread). One shape
cannot necessarily be better than another shape.
How do the annual energy costs (in dollars) compare for refrigerators
with top freezers and refrigerators with bottom freezers? The data
below is from the May 2010 issue of Consumer Reports.
• Shape: The distribution for bottom freezers looks skewed right and possibly
bimodal (modes near $58 and $70 per year). The distribution for top
freezers looks roughly symmetric, with its main peak centered near $55.
• Outliers: There appear to be two bottom freezers with unusually high
energy costs (over $140). There are no outliers for the top freezers.
• Center: The typical energy cost for bottom freezers is greater than the
typical cost for the top freezers (midpoint of $69 vs midpoint of $56).
• Spread: There is much more variability in the energy costs for bottom
freezers.
What is the most important thing to remember
when making a stemplot?
• Stemplots (aka stem-and-leaf plots) are simple
graphical displays for fairly small data sets.
• Stemplots give us a quick picture of the
distribution while including the actual
numerical values.
• Just like with all displays, it is important to
remember the LABELS (and a key)!!!!
How to Make a Stemplot
1)Separate each observation into a stem (all but the final
digit) and a leaf (the final digit).
2)Write all possible stems from the smallest to the largest in a
vertical column and draw a vertical line to the right of the
column.
3)Write each leaf in the row to the right of its stem.
4)Arrange the leaves in increasing order out from the stem.
5)Provide a key that explains in context what the stems and
leaves represent.
• Stemplots (Stem-and-Leaf Plots)
– These data represent the responses of 20 female AP Statistics
students to the question, “How many pairs of shoes do you
have?” Construct a stemplot.
50
26
26
31
57
19
24
22
23
38
13
50
13
34
23
30
49
13
15
51
1
1 93335
1 33359
2
2 664233
2 233466
3
3 1840
3 0148
4
4 9
4 9
5
5 0701
5 0017
Stems
Add leaves
Order leaves
Key: 4|9
represents a
female student
who reported
having 49
pairs of shoes.
Add a key
Sometimes it may be beneficial to split stems, which is a method for
spreading out a stemplot that has too few stems (the data tends to be
bunched up). [every number 0-4 goes in the first stem, 5-9 in the second]
Example: Which gender is taller, males or females? A sample of 14year-olds from the UK was randomly selected using the
CensusAtSchool website. Here are the heights of the students (in
cm). Make a back-to-back stemplot and compare the distributions.
Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151, 175,
174, 165, 165, 183, 180
Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157, 158, 153,
161, 165, 165, 159, 168, 153, 166, 158, 158, 166
If we opted to not split stems:
By
splitting
stems:
Male: 154, 157, 187, 163, 167, 159, 169, 162, 176, 177, 151,
175, 174, 165, 165, 183, 180
Female: 160, 169, 152, 167, 164, 163, 160, 163, 169, 157,
158, 153, 161, 165, 165, 159, 168, 153, 166, 158, 158, 166
Shape: The female distribution
is skewed left unimodal. The
male distribution is symmetric
unimodal.
Outliers: Neither distribution
appears to contain outliers.
Centers: The males have a
larger center than the females
(median of 167 centimeters vs
median of 162 centimeters [avg
the middle two].
Spread: The male distribution
has greater variability than the
female distribution.
• Histograms
– Quantitative variables often take many values. A graph of the
distribution may be clearer if nearby values are grouped
together.
– The most common graph of the distribution of one quantitative
variable is a histogram.
How to Make a Histogram
1)Divide the range of data into classes of equal width.
2)Find the count (frequency) or percent (relative frequency) of
individuals in each class.
3)Label and scale your axes and draw the histogram. The
height of the bar equals its frequency. Adjacent bars should
touch, unless a class contains no individuals.
It might make it easier to visualize a dotplot first before creating a
histogram.
• Here are the run totals for the 30 MLB teams in 2008. Note: the
Astros were still in the NL.
• Here is a dotplot showing the runs scored for
the 30 MLB teams.
Step 1: Divide the data into 5 to 10 equally wide classes.
For this example, we can use classes that are 50 runs wide.
Therefore, our first class will be 600-650 runs, the next class is 650700 runs, etc…
Step 2: Count how many observations are in
each class. If an observation falls exactly on a
border line, it is considered part of the class
above the boundary. For example, the
observation on 750 would count as part of the
750-800 class.
Step 3: Label and scale your axes and draw the
histogram. The height of the bar equals its
frequency. Adjacent bars should touch, unless a
class contains no individuals.
• The smallest
observation is 93.2
and the largest is 106.1
We could choose
classes of width 2
starting at 93.
How do you make a histogram?
• Already answered!
• Points of emphasis: labels, equal class widths,
make sure values that fall on the boundaries
go in the class above.
Why would we prefer a relative frequency
histogram to a frequency histogram?
• When comparing distributions of different
sample size! When the number of
observations are not equal, a fair comparison
cannot be made using just the frequency.
Think of the radio stations in NJ vs radio
stations in the USA example.
• Using Histograms Wisely
– Here are several cautions based on common mistakes students
make when using histograms.
Cautions
1)Don’t confuse histograms and bar graphs.
2)Don’t use counts (in a frequency table) or percents (in a
relative frequency table) as data.
3)Use percents instead of counts on the vertical axis when
comparing distributions with different numbers of
observations.
4)Just because a graph looks nice, it’s not necessarily a
meaningful display of data.
• Follow pages 36-37 to make a histogram on
the TI-84!
• Note:
– To change the boundaries, press WINDOW.
– Xmin defines where the first class begins and Xscl
defines the class width.
– Xmax, Ymin, and Ymax define how big the window
will be.