Describing Data:
Displaying and Exploring Data
Unit 1: One Variable Statistics
CCSS: N-Q (1-3); S-ID 1
McGraw-Hill/Irwin
©The McGraw-Hill Companies, Inc. 2008
CCSS:
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Domain: Quantities N-Q
Cluster: Reason quantitatively and use units to solve problems
Standard:
N.Q.1 Use units as a way to understand problems and to guide the
solution of multi‐step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and the
origin in graphs and data displays.
N.Q.2 Define appropriate quantities for the purpose of descriptive
modeling.
N.Q.3 Choose a level of accuracy appropriate to limitations on
measurement when reporting quantities.
CCSS
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Domain: Interpreting Categorical & Quantitative Data S-ID
 Cluster: Summarize, represent, and interpret data on a
single count or measurement variable
Standard:
S-ID.1.Represent data with plots on the real number line (dot
plots, histograms, and box plots).
S-ID.2.Use statistics appropriate to the shape of the data
distribution to compare center (median, mean) and spread
(interquartile range,
standard deviation) of two or more different data sets.
S-ID.3. Interpret differences in shape, center, and spread in
the
context of the data sets, accounting for possible effects of
extreme data points (outliers).
Mathematical Practice
1. Make sense of problems, and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments, and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for, and make use of, structure.
8. Look for, and express regularity in, repeated
reasoning.
GOALS
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Develop and interpret a dot plot.
Develop and interpret a stem-and-leaf display.
Compute and understand quartiles
Compute and understand the coefficient of
skewness.
Draw and interpret a scatter diagram.
Construct and interpret a contingency table.
Dot Plots
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A dot plot groups the data as little as possible and
the identity of an individual observation is not lost.
To develop a dot plot, each observation is simply
displayed as a dot along a horizontal number line
indicating the possible values of the data.
If there are identical observations or the
observations are too close to be shown individually,
the dots are “piled” on top of each other.
Dot Plots - Examples
Reported below are the number of
vehicles sold in the last 24 months at
Smith Ford Mercury Jeep, Inc., in Kane,
Pennsylvania, and Brophy Honda
Volkswagen in Greenville, Ohio.
Construct dot plots, box plots, and
stem- and-leaf plot, report summary
statistics for the two small-town Auto
USA lots.
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Dot Plot – Minitab Example
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Stem-and-Leaf
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In Chapter 2, we showed how to organize data into a frequency
distribution. The major advantage to organizing the data into a
frequency distribution is that we get a quick visual picture of the
shape of the distribution.
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One technique that is used to display quantitative information in
a condensed form is the stem-and-leaf display.
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Stem-and-leaf display is a statistical technique to present a
set of data. Each numerical value is divided into two parts. The
leading digit(s) becomes the stem and the trailing digit the leaf.
The stems are located along the vertical axis, and the leaf
values are stacked against each other along the horizontal axis.
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Advantage of the stem-and-leaf display over a frequency
distribution - the identity of each observation is not lost.
Stem-and-Leaf – Example
Suppose the seven observations in
the 90 up to 100 class are: 96, 94,
93, 94, 95, 96, and 97.
The stem value is the leading digit or
digits, in this case 9. The leaves
are the trailing digits. The stem is
placed to the left of a vertical line
and the leaf values to the right.
The values in the 90 up to 100
class would appear as
Then, we sort the values within each
stem from smallest to largest.
Thus, the second row of the stemand-leaf display would appear as
follows:
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Another Measure of Variability
The Interquartile Range (IQR) of a numeric
data set is the difference between the third and
first quartiles.
IQR = L75 – L25
This measure of variability tells the range of
values of the middle 50% of the data set.
The first and third quartiles are sometimes
denoted by Q1 = L25 and Q3 = L75.
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Example: ATM Usage
Let’s look at the spread of the data on Aloha
ATM usage using both standard deviation and
quartiles. If we use MegaStat, and choose
Descriptive Statistics, we may find the quartiles
by specifying the Minimum, Maximum, Range
and the Median, Quartiles, Mode, Outliers
options. Or we may do a stem-and-leaf plot
and locate the quartiles on the graph.
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Boxplots – Graphical Use of the 5Number Summary
The 5-number summary of a data set consists
of the minimum value, the first, second
(median) and third quartiles, and the maximum
value of the data. We can use these five
numbers to construct a simple graph, called a
boxplot, of the data. We will compare the
characteristics of the boxplot with those of the
stem-and-leaf plot for several data sets, including the
Whitner Autoplex data and the radio spot data. But
first, another example.
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Boxplot - Example
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Boxplot Example
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Skewness
Another characteristic of a set of data is
the shape.
 There are four shapes commonly
observed:
– symmetric,
– positively skewed,
– negatively skewed,
– bimodal.
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Skewness - Formulas for Computing
The coefficient of skewness can range from -3 up to 3.
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A value near -3, such as -2.57, indicates considerable negative skewness.
A value such as 1.63 indicates moderate positive skewness.
A value of 0, which will occur when the mean and median are equal,
indicates the distribution is symmetrical and that there is no skewness
present.
Commonly Observed Shapes
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Describing Relationship between Two
Variables
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One graphical technique we
use to show the relationship
between variables is called a
scatter diagram.
To draw a scatter diagram we
need two variables. We scale
one variable along the
horizontal axis (X-axis) of a
graph and the other variable
along the vertical axis (Yaxis).
Describing Relationship between Two
Variables – Scatter Diagram Examples
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