1. Chapter 2 Descriptive Statistics: Tabular and Graphical Methods
Summarizing the Qualitative Data
Frequency Distribution
Relative Frequency
Percent Frequency Distribution
Bar Graph
Pie Chart

2. Frequency Distribution
A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several classes.

3. Example: Marada Inn
Guests staying at Marada Inn were asked to rate the
quality of their accommodations as being excellent,
above average, average, below average, or poor. The
ratings provided by a sample of 20 quests are shown
below.
Below Average Average Above Average
Above Average Above Average Above Average Above Average Below Average Below Average Average Poor Poor
Above Average Excellent Above Average
Average Above Average Average
Above Average Average

4. Example: Marada Inn Frequency Distribution Rating Frequency Poor 2
Below Average
Average
Above Average
Excellent
Total

5. Relative Frequency Distribution
The relative frequency of a class is the fraction or proportion of the total number of data items belonging to the class.
A relative frequency distribution is a tabular summary of a set of data showing the relative frequency for each class.

6. Percent Frequency Distribution
The percent frequency of a class is the relative frequency multiplied by 100.
A percent frequency distribution is a tabular summary of a set of data showing the percent frequency for each class.

7. Example: Marada Inn
Relative Frequency and Percent Frequency Distributions
Relative Percent
Rating Frequency Frequency
Poor
Below Average
Average
Above Average
Excellent
Total

8. Bar Graph
A bar graph is a graphical device for depicting qualitative data.
On the horizontal axis we specify the labels that are used for each of the classes.
A frequency, relative frequency, or percent frequency scale can be used for the vertical axis.
The bars are separated to emphasize the fact that each class is a separate category.

9. Example: Marada Inn Bar Graph 1 2 3 4 5 6 7 8 9 Poor Below Average
Above
Excellent
Frequency
Rating

10. Pie Chart
The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data.
First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class.
Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) =
90 degrees of the circle.

11. Example: Marada Inn Pie Chart Average 25% Below 15% Poor 10% Above 45%
Exc. 5%
Quality Ratings

12. Example: Marada Inn Insights Gained from the Preceding Pie Chart
One-half of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager.
For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager.

13. Summarizing Quantitative Data
Frequency Distribution
Relative Frequency
Percent Frequency Distributions
Cumulative Distributions
Dot Plot
Histogram
Ogive/ Frequency Polygon

14. Example: Hudson Auto Repair
The manager of Hudson Auto would like to get a
better picture of the distribution of costs for engine
tune-up parts. A sample of 50 customer invoices has
been taken and the costs of parts, rounded to the
nearest dollar, are listed below.

15. Frequency Distribution
Guidelines for Selecting Number of Classes
Use between 5 and 20 classes.
Data sets with a larger number of elements usually require a larger number of classes.
Smaller data sets usually require fewer classes.

16. Frequency Distribution
Guidelines for Selecting Width of Classes
Use classes of equal width.
Approximate Class Width =

17. Example: Hudson Auto Repair
Frequency Distribution
If we choose six classes:
Approximate Class Width = ( )/6 = 9.5 10
Cost ($) Frequency
Total

18. Example: Hudson Auto Repair
Relative Frequency and Percent Frequency Distributions
Relative Percent
Cost ($) Frequency Frequency
Total

19. Example: Hudson Auto Repair
Insights Gained from the Percent Frequency Distribution
Only 4% of the parts costs are in the $50-59 class.
30% of the parts costs are under $70.
The greatest percentage (32% or almost one-third) of the parts costs are in the $70-79 class.
10% of the parts costs are $100 or more.

20. Dot Plot
One of the simplest graphical summaries of quantitative data is a dot plot.
A horizontal axis shows the range of data values.
Then each data value is represented by a dot placed above the axis.

21. Example: Hudson Auto Repair
Dot Plot .
Cost ($)

22. Histogram
Another common graphical presentation of quantitative data is a histogram.
The variable of interest is placed on the horizontal axis.
A rectangle is drawn above each class interval’s frequency, relative frequency, or percent frequency.
Unlike a bar graph, a histogram has no natural separation between rectangles of classes.

23. Example: Hudson Auto Repair
Histogram 18 16 14 12
Frequency 10 8 6 4 2
Parts
Cost ($)

24. Cumulative Distributions
Cumulative frequency distribution -- shows the number of items with values less than or equal to the upper limit of each class.
Cumulative relative frequency distribution -- shows the proportion of items with values less than or equal to the upper limit of each class.
Cumulative percent frequency distribution -- shows the percentage of items with values less than or equal to the upper limit of each class.

25. Example: Hudson Auto Repair
Cumulative Distributions
Cumulative Cumulative
Cumulative Relative Percent
Cost ($) Frequency Frequency Frequency
<
<
<
<
<
<

26. Ogive An ogive is a graph of a cumulative distribution.
The data values are shown on the horizontal axis.
Shown on the vertical axis are the:
cumulative frequencies, or cumulative relative frequencies, or cumulative percent frequencies
The frequency (one of the above) of each class is plotted as a point.
The plotted points are connected by straight lines.

27. Example: Hudson Auto Repair
Ogive
Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit gaps from 59 to 60, 69 to 70, and so on.
These gaps are eliminated by plotting points halfway between the class limits.
Thus, 59.5 is used for the class, 69.5 is used for the class, and so on.

28. Example: Hudson Auto Repair
Ogive with Cumulative Percent Frequencies
100 80 60
Cumulative Percent Frequency 40 20
Parts
Cost ($)