1. Frequency Distribution
Intro to Statistics:
Frequency Distribution
And Graphs
Ernesto Diaz
Professor of Mathematics 1

2. Frequency Distributions and Graphs 14.1
Copyright © Cengage Learning. All rights reserved.

3. Frequency Distribution

4. Frequency Distribution
The difference between the lower limit of one class and the lower limit of the next class is called the interval of the class. After determining the number of values within a class, termed the frequency.
The end result of this classification and tabulation is called a frequency distribution.
For example, suppose that you roll a pair of dice 50 times and obtain these outcomes: 3, 2, 6, 5, 3, 8, 8, 7, 10, 9, 7, 5, 12, 9, 6, 11, 8, 11, 11, 8, 7,
7, 7, 10, 11, 6, 4, 8, 8, 7, 6, 4, 10, 7, 9, 7, 9, 6, 6, 9, 4, 4, 6,
3, 4, 10, 6, 9, 6, 11

5. Frequency Distribution
We can organize these data in a convenient way by using a frequency distribution, as shown in Table 14.1.
Frequency Distribution for 50 Rolls of a Pair of Dice
Table 14.1

6. Example 1 – Make a frequency diagram
Make a frequency distribution for the high temperatures (in °F) for October on Lake Erie as summarized in Table 14.2.
High Temperatures on Lake Erie
for the Month of October
Table 14.2

7. Example 1 – Solution
First, notice that the included temperatures range from a low of 61 to a high of 86.
If we were to list each temperature and make tally marks for each, we would have a long table with few entries for each temperature.
Instead, we arbitrarily divide the temperatures into eight categories. Make the tallies and total the number in each group.
The result is called a grouped frequency distribution.

8. Frequency Distribution
A graphical method useful for organizing large sets of data is a stem-and-leaf plot. Consider the data shown in Table 14.3.
Best Actors, 1928–2009
Table 14.3

9. Frequency Distribution
cont’d
Best Actors, 1928–2009
Table 14.3

10. Example 2 – Make a stem-and-leaf plot
Construct a stem-and-leaf plot for the best actor ages in Table 14.3.
Solution:
Stem-and-leaf plot of ages of best actor, 1928–2009.
This plot is useful because it is easy to see that most best actor winners received the award in their forties.

11. Frequency Distribution
To help us understand the relationship between and among variables, we use a diagram called a graph.
In this section, we consider bar graphs, line graphs, circle graphs, and pictographs.

12. Bar Graphs

13. Bar Graphs
A bar graph compares several related pieces of data using horizontal or vertical bars of uniform width. There must be some sort of scale or measurement on both the horizontal and vertical axes.
An example of a bar graph is shown in Figure 14.1, which shows the data from Table 14.1.
Outcomes of experiment of rolling a pair of dice
Figure 14.1

14. Example 3 – Make a bar graph
Construct a bar graph for the Lake Erie temperatures given in Example 1. Use the grouped categories.
Solution: To construct a bar graph, draw and label the horizontal and vertical axes, as shown in part a of Figure 14.2.
Draw and label axes and scales.
Bar graph for Lake Erie temperatures
Figure 14.2(a)

15. Example 3 – Solution
cont’d
It is helpful (although not necessary) to use graph paper. Next, draw marks indicating the frequency, as shown in part b of Figure Finally, complete the bars and shade them as shown in part c.
Mark the frequency levels.
Complete and shade the bars.
Bar graph for Lake Erie temperatures
Bar graph for Lake Erie temperatures
Figure 14.2(b)
Figure 14.2(c)

16. Example 4 – Read a bar graph
Refer to Figure 14.3 to answer the following questions.
Share of U.S. employment in agriculture, manufacturing, and services from 1849 to 2049
Figure 14.3

17. Example 4 – Read a bar graph
cont’d
a. What was the share of U.S. employment in manufacturing in 1999?
b. In which year(s) was U.S. employment in services approximately equal to employment in manufacturing?
c. From the graph, form a conclusion about U.S. employment in agriculture for the period 1849–2049.

18. Example 4 – Solution
a. We see the bar representing manufacturing in 1999 is approximately 20%. (Use a straightedge for help.)
b. Employments in services and manufacturing were approximately equal in 1949.
c. Conclusions, of course, might vary, but one obvious conclusion is that there has been a dramatic decline of employment in agriculture in the United States over this period of time.

19. Line Graphs

20. Line Graphs
A graph that uses a broken line to illustrate how one quantity changes with respect to another is called a line graph.
A line graph is one of the most widely used kinds of graph.

21. Example 5 – Make a line graph
Draw a line graph for the data given in Example 1. Use the previously grouped categories.
Solution
The line graph uses points instead of bars to designate the locations of the frequencies. These points are then connected by line segments.
To plot the points, use the frequency distribution to find the midpoint of each category (62 is the midpoint of the 60–64 category, for example); then plot a point showing the frequency (3, in this example).

22. Example 5 – Solution
cont’d
This step is shown in part a of Figure The last step is to connect the dots with line segments, as shown in part b.
a. Plot the points to represent frequency levels.
b. Connect the dots with line segments.
Constructing a line graph for Lake Erie temperatures
Figure 14.4

23. Circle Graphs

24. Circle Graphs
Another type of commonly used graph is the circle graph, also known as a pie chart.
To create a circle graph, first express the number in each category as a percentage of the total. Then convert this percentage to an angle in a circle.
Remember that a circle is divided into 360°, so we multiply the percent by 360 to find the number of degrees for each category.
You can use a protractor to construct a circle graph, as shown in the next example.

25. Example 7 – Make a circle graph
The 2010 expenses for Karlin Enterprises are shown in Figure Construct a circle graph showing the expenses for Karlin Enterprises.
Karlin Enterprises expenses
Figure 14.6

26. Example 7 – Solution
The first step in constructing a circle graph is to write the ratio of each entry to the total of the entries, as a percent.
This is done by finding the total ($120,000) and then dividing each entry by that total:
Salaries: Rents:
Utilities: Advertising:

27. Example 7 – Solution Shrinkage: Depreciation: Materials/supplies:
cont’d
Shrinkage: Depreciation:
Materials/supplies:

28. Example 7 – Solution
cont’d
A circle has 360°, so the next step is to multiply each percent by 360°:
Salaries: 360°  0.60 = 216°
Advertising: 360°  0.10 = 36°
Rents: 360°  0.20 = 72°
Shrinkage: 360°  0.01 = 3.6°
Materials/supplies: 360°  0.01 = 3.6°
Utilities: 360°  0.05 = 18°
Depreciation: 360°  0.03 = 10.8°

29. Example 7 – Solution
cont’d
Finally, use a protractor to construct the circle graph as shown in Figure 14.7.
Circle graph showing the expenses for Karlin Enterprises
Figure 14.7

30. Pictographs

31. Pictographs
A pictograph is a representation of data that uses pictures to show quantity. Consider the raw data shown in Table 14.4.
A pictograph uses a picture to illustrate data; it is normally used only in popular publications, rather than for scientific applications.
Marital Status (for persons age 65 and older)
Table 14.4

32. Pictographs
For the data in Table 14.4, suppose that we draw pictures of a woman and a man so that each picture represents 1 million persons, as shown in Figure 14.8.
Pictograph showing the marital status of persons age 65 and older
Figure 14.8

33. Statistics
Statistics is the art and science of gathering, analyzing, and making inferences from numerical information (data) obtained in an experiment.
Statistics are divided into two main braches.
Descriptive statistics is concerned with the collection, organization, and analysis of data.
Inferential statistics is concerned with the making of generalizations or predictions of the data collected.

34. Statisticians
A statistician’s interest lies in drawing conclusions about possible outcomes through observations of only a few particular events.
The population consists of all items or people of interest.
The sample includes some of the items in the population.
When a statistician draws a conclusion from a sample, there is always the possibility that the conclusion is incorrect.

35. The Misuses of Statistics

36. Misuses of Statistics
Many individuals, businesses, and advertising firms misuse statistics to their own advantage.
When examining statistical information consider the following:
Was the sample used to gather the statistical data unbiased and of sufficient size?
Is the statistical statement ambiguous, could it be interpreted in more than one way?

37. Example: Misleading Statistics
An advertisement says, “Fly Speedway Airlines and Save 20%”.
Here there is not enough information given.
The “Save 20%” could be off the original ticket price, the ticket price when you buy two tickets or of another airline’s ticket price.
A helped wanted ad read,” Salesperson wanted for Ryan’s Furniture Store. Average Salary: $32,000.”
The word “average” can be very misleading.
If most of the salespeople earn $20,000 to $25,000 and the owner earns $76,000, this “average salary” is not a fair representation.

38. Charts and Graphs
Charts and graphs can also be misleading.
Even though the data is displayed correctly, adjusting the vertical scale of a graph can give a different impression.
A circle graph can be misleading if the sum of the parts of the graphs do not add up to 100%.

39. Example: Misleading Graphs
While each graph presents identical information, the vertical scales have been altered.

40. Example: Misleading Graphs
To correctly interpret a graph, you must analyze the numerical information given in the graph, so as not to be misled by the graph’s shape.

41. Example: Pictographs
Part (b) is designed to exaggerate the difference by increasing each dimension in proportion to the actual amounts of oil consumption.

42. Compare Apples and Oranges
Two doctors in Anytown, CA…
Doctor #1 Doctor #2
What if the populations served by each doctor were very different?
Doctor of the Year?
2/1000
20/1000

43. Number of Crimes Period 1: 76 Period 2: 51 Period 3: 91 Period 4: 76
Data snapshots…
Crime in Anytown, CA…
Number of Crimes
Period 1: 76
Period 2: 51
Period 3: 91
Period 4: 76

44. Unrepresentative findings…
Survey of people in Anytown, CA…
90% of respondents stated that they support using tax dollars to build a new football stadium.
The implication of the above finding is that there is overwhelming support for the stadium…
But what if you were then told that respondents had been sampled from a list of season football ticket holders?

45. Logical “Flipping”… Headline in The Anytown Chronicle:
60% of violent crimes are committed by men who did not graduate from high school.
“Flip”
60% of male high school drop-outs commit violent crimes?

46. But the original and flipped statements have very different meanings!
“One study in Washington State found that 75 percent of a sample of neglect cases involved families with incomes under $10,000.”
In reading statistics such as the above, there is a tendency to want to directionally “Flip” the interpretation
But the original and flipped statements have very different meanings!
75% of neglect cases involved families with incomes under $10,000
DOES NOT MEAN
75% of families with incomes under $10,000 have open neglect cases
Put more simply, just because most neglected children are poor does not mean that most poor children are neglected

47. False Causality…
A study of Anytown residents makes the following claim:
Adults with short hair are, on average, more than 3 inches taller than those with long hair.
Finding an association between two factors does not mean that one causes the other…
Hair Length
Height
Gender X

48. prematurity maltreatment a third factor (Drug use?)
“A number of child characteristics have previously been shown to be associated with risk of maltreatment. Prematurity or low birth weight is frequently reported…”
Should one conclude that prematurity is a causal factor in maltreatment?
prematurity
maltreatment
a third factor
(Drug use?)

49. Misuses of Graphs

50. Misuses of Graphs
The most misused type of graph is the pictograph. Consider the data from Table 14.4.
Such data can be used to determine the height of a three-dimensional object.
Marital Status (for persons age 65 and older)
Table 14.4

51. Misuses of Graphs
When an object (such as a person) is viewed as three-dimensional, differences seem much larger than they actually are. Look at the Figure 14.9 and notice that, as the height and width are doubled, the volume is actually increased eightfold.
Doubling height and width increases volume 8-fold
Examples of misuses in pictographs
Figure 14.9

52. Misuses of Graphs
This pictograph fallacy carries over to graphs of all kinds, especially since software programs easily change two dimensional scales to three-dimensional scales.
For example, percentages are represented as heights on the scale shown in Figure 14.10a, but the software has incorrectly drawn the graph as three-dimensional bars.
Heights are used to graph three-dimensional objects.
Misuse of graphs
Figure 14.10(a)

53. Misuses of Graphs
Another fallacy is to choose the scales to exaggerate or diminish real differences. Even worse, graphs are sometimes presented with no scale whatsoever, as illustrated in a graphical “comparison” between Anacin and “regular strength aspirin” shown in Figure 14.10b.
No scale is shown in this Anacin advertisement
Misuse of graphs
Figure 14.10(b)