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1. Elementary Statistics
Chapter 2: Exploring
Data with Tables and
Graphs
2.3 Graphs that
Enlighten and Graphs
that Deceive
1

2. Chapter 2:
Exploring Data with Tables and Graphs
2.1 Frequency Distributions for Organizing and
Summarizing Data
2.2 Histograms
2.3 Graphs that Enlighten and Graphs that Deceive
2.4 Scatterplots, Correlation, and Regression
2
Objectives:
1. Organize data using a frequency distribution.
2. Represent data in frequency distributions graphically using histograms, frequency polygons, and ogives.
3. Represent data using bar graphs, Pareto charts, time series graphs, and pie graphs.
4. Draw and interpret a stem and leaf plot.
5. Draw and interpret a scatter plot for a set of paired data.

3. Key Concept:
Common graphs that foster understanding of data and some graphs that are
deceptive because they create impressions about data that are somehow misleading
or wrong.
2.3 Graphs that Enlighten and Graphs that Deceive
3
Dot plots & its Features:
A graph of quantitative data in which each data value is plotted as a point (or dot) above a horizontal
scale of values. Dots representing equal values are stacked.
Displays the shape of distribution of data.
It is usually possible to recreate the original list of data values.
Example 1: Construct a Dotplot and find an outlier if any.
Number 36 seems to be an outlier.
80 94 58 66 56 82 78 86 88 56 36 66 84 76 78 64 66 78 60 64

4. Stemplots (or stem-and-leaf plot)
Represents quantitative data by separating each value into two parts: the
stem (such as the leftmost digit) and the leaf (such as the rightmost digit).
Features of a Stemplot
Shows the shape of the distribution of the data.
Retains the original data values.
The sample data are sorted (arranged in order).
4
Graphs that Enlighten: Stemplots (or stem-and-leaf plot)
Example 2:
Construct a Stemplot.
80 94 58 66 56 82 78 86 88 56 36 66 84 76 78 64 66 78 60 64

5. Time-Series Graph
A graph of time-series data, which are quantitative data that have been collected at different
points in time, such as monthly or yearly
Feature of a Time-Series Graph: Reveals information about trends over time.
5
Graphs that Enlighten
From 1990 to 2010, there seems to be upward trend
that appears to be leveling off in recent years. It would
help close the gender gap in earning.
Example 3: Construct a Time-Series Graph.
Women’s median earning as a percentage of men’s median earning from 1990 to 2010
are listed as follows; Is there a trend? Any affect?

6. Example 4: Construct the Pareto Chart.
A study of retractions (admission of wrong claim) in biomedicine
showed, 472 were due to error, 225 were due to plagiarism, 881 were due
to fraud, 295 were due to duplications of publications, and 249 had other
causes. Construct a Pareto chart. Among such retractions, does
misconduct (fraud, duplication, plagiarism) appear to be a major factor?
Bar Graphs: A graph of bars of equal width to show frequencies of categories of categorical
(or qualitative) data. The bars may or may not be separated by small gaps.
Feature of a Bar Graph: Shows the relative distribution of categorical data so that it is easier
to compare the different categories.
6
Graphs that Enlighten
Pareto Charts: A Pareto chart is a bar graph for categorical data, with the
added condition that the bars are arranged in descending order according
to frequencies, so the bars decrease in height from left to right.
Features of a Pareto Chart: Shows the relative distribution of categorical data
so that it is easier to compare the different categories.
Draws attention to the more important categories.
Yes, it
does.

7. Pie Charts
A very common graph that depicts categorical data as slices of a
circle, in which the size of each slice is proportional to the
frequency count for the category
Feature of a Pie Chart: Each sector =→ 𝟑𝟔𝟎𝟎
× %
Shows the distribution of categorical data in a commonly used format.
7
Graphs that Enlighten
Example 5: Construct the Pie Chart.
A study of retractions (admission of wrong claim) in biomedicine
showed, 472 were due to error, 225 were due to plagiarism, 881 were due
to fraud, 295 were due to duplications of publications, and 249 had other
causes. Construct a Pareto chart. Among such retractions, does
misconduct (fraud, duplication, plagiarism) appear to be a major factor?

8. Frequency Polygon
A graph using line segments connected to points located directly above class midpoint values
A frequency polygon is very similar to a histogram, but a frequency polygon uses line segments instead of bars.
Relative Frequency Polygon
A variation of the basic frequency polygon is the relative frequency polygon, which uses relative frequencies (proportions
or percentages) for the vertical scale.
8
Graphs that Enlighten

9. Relative F
f/n
9
Graphs that Enlighten
Example 6
Construct a histogram, frequency polygon, and ogive using relative
frequencies for the distribution (shown here) of the miles that 20
randomly selected runners ran during a given week.
Class
Boundaries
F
5.5 - 10.5
10.5 - 15.5
15.5 - 20.5
20.5 - 25.5
25.5 - 30.5
30.5 - 35.5
35.5 - 40.5
1
2
3
5
4
3
2
1/20 =
2/20 =
3/20 =
5/20 =
4/20 =
3/20 =
2/20 =
f = 20 rf = 1.00
0.05
0.10
0.15
0.25
0.20
0.15
0.10
Midpoints
8
13
18
23
28
33
38

10. Relative F
f/n
10
Graphs that Enlighten
Example 6 Continued
Construct a histogram, frequency polygon, and ogive using
relative frequencies for the distribution (shown here) of the miles
that 20 randomly selected runners ran during a given week.
Class
Boundaries
F
5.5 - 10.5
10.5 - 15.5
15.5 - 20.5
20.5 - 25.5
25.5 - 30.5
30.5 - 35.5
35.5 - 40.5
1
2
3
5
4
3
2
1/20 =
2/20 =
3/20 =
5/20 =
4/20 =
3/20 =
2/20 =
f = 20 rf = 1.00
0.05
0.10
0.15
0.25
0.20
0.15
0.10
Midpoints
8
13
18
23
28
33
38

11. 11
Graphs that Enlighten
Example 6
Construct an ogive:
It’s a line graph that depicts cumulative frequencies
Class
Boundaries
F
5.5 - 10.5
10.5 - 15.5
15.5 - 20.5
20.5 - 25.5
25.5 - 30.5
30.5 - 35.5
35.5 - 40.5
1
2
3
5
4
3
2
1/20 =
2/20 =
3/20 =
5/20 =
4/20 =
3/20 =
2/20 =
f = 20 rf = 1.00
0.05
0.10
0.15
0.25
0.20
0.15
0.10
Relative F
f/n
Class
Boundaries
Cumulative F
5.5 - 10.5
5.5 - 15.5
5.5 - 20.5
5.5 - 25.5
5.5 - 30.5
5.5 - 35.5
5.5 - 40.5
1
3
6
11
15
18
20
Cum. Rel.
Frequency
1/20 =
3/20 =
6/20 =
11/20 =
15/20 =
18/20 =
20/20 =
0.05
0.15
0.30
0.55
0.75
0.90
1.00

12. 12
Graphs that Enlighten
Example 6
Construct an ogive:
It’s a line graph that depicts cumulative frequencies
Ogives use upper class boundaries and cumulative frequencies or
cumulative relative frequencies of the classes.
Class
Boundaries
Cumulative F
5.5 - 10.5
5.5 - 15.5
5.5 - 20.5
5.5 - 25.5
5.5 - 30.5
5.5 - 35.5
5.5 - 40.5
1
3
6
11
15
18
20
Cum. Rel.
Frequency
1/20 =
3/20 =
6/20 =
11/20 =
15/20 =
18/20 =
20/20 =
0.05
0.15
0.30
0.55
0.75
0.90
1.00 What percent of runners
ran 22 miles or less? About 41%

13. Nonzero Vertical Axis
A common deceptive graph involves using a vertical scale that starts at some value
greater than zero to exaggerate differences between groups.
13
Graphs that Deceive
Always examine a graph carefully to see whether a vertical axis begins at
some point other than zero so that differences are exaggerated.

14. Drawings of objects, called pictographs, are often misleading. Data that are one-dimensional in
nature (such as budget amounts) are often depicted with two-dimensional objects (such as
dollar bills) or three-dimensional objects (such as stacks of coins, homes, or barrels).
By using pictographs, artists can create false impressions that grossly distort differences by
using these simple principles of basic geometry:
When you double each side of a square, its area doesn’t merely double; it increases by a factor of four.
When you double each side of a cube, its volume doesn’t merely double; it increases by a factor of eight.
14
Graphs that Deceive: Pictographs
Misleading. Depicts one-dimensional
data with three-dimensional boxes. Last
box is 64 times as large as first box, but
income is only 4 times as large. Fair, objective,
clear from any
distracting features.