1. Section 13.3 Frequency Distribution and Statistical Graphs

2. What You Will Learn Frequency Distributions Histograms
Frequency Polygons
Stem-and-Leaf Displays
Circle Graphs

3. Frequency Distribution
A piece of data is a single response to an experiment.
A frequency distribution is a listing of observed values and the corresponding frequency of occurrence of each value.

4. Example 1: Frequency Distribution
The number of children per family is recorded for 64 families surveyed. Construct a frequency distribution of the following data:

5. Example 1: Frequency Distribution

7. Example 1: Frequency Distribution
Eight families had no children,
11 families had one child,
18 families had two children,
and so on.
Note that the sum of the frequencies is equal to the original number of pieces of data, 64.

8. Rules for Data Grouped by Classes
1. The classes should be of the same “width.”
2. The classes should not overlap.
3. Each piece of data should belong to only one class.
Often suggested that there be 5 – 12 classes.

9. Definitions
Midpoint of a class is found by adding the lower and upper class limits and dividing the sum by 2.

10. Example 3: A Frequency Distribution of Family Income
The following set of data represents the family income (in thousands of dollars, rounded to the nearest hundred) of 15 randomly selected families.
50.7
56.3
39.8
40.3
35.5
48.8
53.7
44.6
52.4
65.2
40.9
44.7
45.8
31.8
46.5

11. Example 3: A Frequency Distribution of Family Income
Construct a frequency distribution with a first class of 31.5–37.6.
Solution
Rearrange data from lowest to highest.
65.2
52.4
46.5
44.6
39.8
56.3
50.7
45.8
40.9
35.5
53.7
48.8
44.7
40.3
31.8

12. Example 3: A Frequency Distribution of Family Income
Solution
Class width is
37.6 – 31.5
= 6.2.

13. Example 3: A Frequency Distribution of Family Income
Solution
The modal class is 43.9–50.0.
The class mark of the first class is ( )÷2 =

14. Histograms
A histogram is a graph with observed values on its horizontal scale and frequencies on its vertical scale.
Because histograms and other bar graphs are easy to interpret visually, they are used a great deal in newspapers and magazines.

15. Constructing a Histogram
A bar is constructed above each observed value (or class when classes are used), indicating the frequency of that value (or class).
The horizontal scale need not start at zero, and the calibrations on the horizontal and vertical scales do not have to be the same.

16. Constructing a Histogram
The vertical scale must start at zero. To accommodate large frequencies on the vertical scale, it may be necessary to break the scale.

17. Example 4: Construct a Histogram
The frequency distribution developed in Example 1 is shown on the next slide. Construct a histogram of this frequency distribution.

19. Example 4: Construct a Histogram
Solution
Vertical scale: 0 – 20.
Horizontal scale: 0 – 9.
Bar above 0 extends to 8.
Above 1, bar extends to 11.
Bar above 2 extends to 18.
Continue this procedure for each observed value to get the histogram on the next slide.

21. Frequency Polygon
Frequency polygons are line graphs with scales the same as those of the histogram; that is, the horizontal scale indicates observed values and the vertical scale indicates frequency.

22. Constructing a Frequency Polygon
Place a dot at the corresponding frequency above each of the observed values.
Then connect the dots with straight-line segments.

23. Constructing a Frequency Polygon
When constructing frequency polygons, always put in two additional class marks, one at the lower end and one at the upper end on the horizontal scale.
Since the frequency at these added class marks is 0, the end points of the frequency polygon will always be on the horizontal scale.

24. Example 5: Construct a Histogram
Construct a frequency polygon of the frequency distribution in Example 1, found on the next slide.

26. Example 5: Construct a Histogram
Solution
Vertical scale: 0 – 20.
Horizontal scale: 0 – 9, plus one at each end.
Place a mark above 0 at 8.
Place a mark above 1 at 11.
And so on.
Connect the dots, bring the end points down to the horizontal axis.

28. Stem-and-Leaf Display
A stem-and-leaf display is a tool that organizes and groups the data while allowing us to see the actual values that make up the data.

29. Constructing a Stem-and-Leaf Display
To construct a stem-and-leaf display each value is represented with two different groups of digits.
The left group of digits is called the stem.
The remaining group of digits on the right is called the leaf.

30. Constructing a Stem-and-Leaf Display
There is no rule for the number of digits to be included in the stem.
Usually the units digit is the leaf and the remaining digits are the stem.

31. Example 8: Constructing a Stem-and-Leaf Display
The table below indicates the ages of a sample of 20 guests who stayed at Captain Fairfield Inn Bed and Breakfast. Construct a stem-and-leaf display.

32. Example 8: Constructing a Stem-and-Leaf Display
Solution
Stem 2 3 4 5 6 7
Leaves
9 7 8
1 9 2
0 2 8
1 2

33. Circle Graphs
Circle graphs (also known as pie charts) are often used to compare parts of one or more components of the whole to the whole.

34. Example 9: Circus Performances
Eight hundred people who attended a Ringling Bros. and Barnum & Bailey Circus were asked to indicate their favorite performance. The circle graph shows the percentage of respondents that answered tigers, elephants, acrobats, jugglers, and other. Determine the number of respondents for each category.

35. Example 9: Circus Performances

36. Example 9: Circus Performances
Solution
To determine the number of respondents in a category, we multiply the percentage for each category, written as a decimal number, by the total number of people, 800.

37. Example 9: Circus Performances
Solution
Tigers 38%
Elephants 26%
Acrobats 17%
Jugglers 14%
Other 5%
Create the table on the next slide.

38. Example 9: Circus Performances
Solution

39. Example 9: Circus Performances
Solution
304 people indicated tigers were their favorite performance,
208 indicated elephants,
136 people indicated the acrobats,
112 people indicated the jugglers, and
40 people indicated some other performance.