1. Organizing Quantitative Data: The Popular Displays
Chapter 2 Section 2
Organizing Quantitative Data:
The Popular Displays

2. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 7 6 1 2 3 5 4

3. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 1 2 3 5 4 7 6

4. Chapter 2 – Section 2
Raw quantitative data comes as a list of values … each value is a measurement, either discrete or continuous
Comparisons (one value being more than or less than another) can be performed on the data values
Mathematical operations (addition, subtraction, …) can be performed on the data values

5. Chapter 2 – Section 2
Discrete quantitative data can be presented in tables in several of the same ways as qualitative data
Values listed in a table
By a frequency table
By a relative frequency table
We use the discrete values instead of the category names

6. Chapter 2 – Section 2 Consider the following data
We would like to compute the frequencies and the relative frequencies

7. Chapter 2 – Section 2
The resulting frequencies and the relative frequencies

8. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 1 2 3 5 4 7 6

9. Chapter 2 – Section 2
Discrete quantitative data can be presented in bar graphs in several of the same ways as qualitative data
We use the discrete values instead of the category names
We arrange the values in ascending order
For discrete data, these are called histograms

10. Chapter 2 – Section 2 Example of histograms for discrete data
Frequencies
Relative frequencies

11. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 1 2 3 5 4 7 6

12. Chapter 2 – Section 2
Continuous data cannot be put directly into frequency tables since they do not have any obvious categories
Categories are created using classes, or intervals of numbers
The continuous data is then put into the classes

13. Chapter 2 – Section 2 For ages of adults, a possible set of classes is
20 – 29
30 – 39
40 – 49
50 – 59
60 and older
For the class 30 – 39
30 is the lower class limit
39 is the upper class limit

14. Chapter 2 – Section 2
The class width is the difference between the upper class limit and the lower class limit
For the class 30 – 39, the class width is
40 – 30 = 10
Why isn’t the class width 39 – 30 = 9?
The class 30 – 39 years old actually is 30 years to 39 years 364 days old … or 30 years to just less than 40 years old
The class width is 10 years, all adults in their 30’s
The class width is the difference between the upper class limit and the lower class limit
For the class 30 – 39, the class width is
40 – 30 = 10

15. Chapter 2 – Section 2
All the classes (20 – 29, 30 – 39, 40 – 49, 50 – 59) all have the same widths, except for the last class
The class “60 and above” is an open-ended class because it has no upper limit
Classes with no lower limits are also called open-ended classes
All the classes (20 – 29, 30 – 39, 40 – 49, 50 – 59) all have the same widths, except for the last class
All the classes (20 – 29, 30 – 39, 40 – 49, 50 – 59) all have the same widths, except for the last class
The class “60 and above” is an open-ended class because it has no upper limit

16. Chapter 2 – Section 2
The classes and the number of values in each can be put into a frequency table
In this table, there are 1147 subjects between 30 and 39 years old
Age
Number
20 – 29
533
30 – 39
1147
40 – 49
1090
50 – 59
493
60 and older
110

17. Chapter 2 – Section 2
Good practices for constructing tables for continuous variables
The classes should not overlap
The classes should not have any gaps between them
The classes should have the same width (except for possible open-ended classes at the extreme low or extreme high ends)
The class boundaries should be “reasonable” numbers
The class width should be a “reasonable” number
Good practices for constructing tables for continuous variables
The classes should not overlap
The classes should not have any gaps between them
The classes should have the same width (except for possible open-ended classes at the extreme low or extreme high ends)
The class boundaries should be “reasonable” numbers
Good practices for constructing tables for continuous variables
The classes should not overlap
The classes should not have any gaps between them
Good practices for constructing tables for continuous variables
The classes should not overlap
Good practices for constructing tables for continuous variables
The classes should not overlap
The classes should not have any gaps between them
The classes should have the same width (except for possible open-ended classes at the extreme low or extreme high ends)

18. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 1 2 3 5 4 7 6

19. Chapter 2 – Section 2
Just as for discrete data, a histogram can be created from the frequency table
Instead of individual data values, the categories are the classes – the intervals of data

20. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 1 2 3 5 4 7 6

21. Chapter 2 – Section 2
A stem-and-leaf plot is a different way to represent data that is similar to a histogram
To draw a stem-and-leaf plot, each data value must be broken up into two components
The stem consists of all the digits except for the right most one
The leaf consists of the right most digit
For the number 173, for example, the stem would be “17” and the leaf would be “3”
A stem-and-leaf plot is a different way to represent data that is similar to a histogram

22. Chapter 2 – Section 2 In the stem-and-leaf plot below
The smallest value is 56
The largest value is 180
The second largest value is 178

23. Chapter 2 – Section 2 To read a stem-and-leaf plot Stem-and-leaf plots
Read the stem first
Attach the leaf as the last digit of the stem
The result is the original data value
Stem-and-leaf plots
Display the same visual patterns as histograms
Contain more information than histograms
Could be more difficult to interpret (including getting a sore neck)
To read a stem-and-leaf plot
Read the stem first
Attach the leaf as the last digit of the stem
The result is the original data value

24. Chapter 2 – Section 2 To draw a stem-and-leaf plot
Write all the values in ascending order
Find the stems and write them vertically in ascending order
For each data value, write its leaf in the row next to its stem
The resulting leaves will also be in ascending order
The list of stems with their corresponding leaves is the stem-and-leaf plot
To draw a stem-and-leaf plot
Write all the values in ascending order
Find the stems and write them vertically in ascending order
For each data value, write its leaf in the row next to its stem
The resulting leaves will also be in ascending order
To draw a stem-and-leaf plot
Write all the values in ascending order
Find the stems and write them vertically in ascending order
To draw a stem-and-leaf plot
Write all the values in ascending order
To draw a stem-and-leaf plot
Write all the values in ascending order
Find the stems and write them vertically in ascending order
For each data value, write its leaf in the row next to its stem

25. Chapter 2 – Section 2 Modifications to stem-and-leaf plots
Sometimes there are too many values with the same stem … we would need to split the stems (such as having in one stem and in another)
If we wanted to compare two sets of data, we could draw two stem-and-leaf plots using the same stem, with leaves going left (for one set of data) and right (for the other set)
There are cases where constructing a descending stem-and-leaf plot could also be appropriate (for test scores, for example)
Modifications to stem-and-leaf plots
Sometimes there are too many values with the same stem … we would need to split the stems (such as having in one stem and in another)
Modifications to stem-and-leaf plots
Sometimes there are too many values with the same stem … we would need to split the stems (such as having in one stem and in another)
If we wanted to compare two sets of data, we could draw two stem-and-leaf plots using the same stem, with leaves going left (for one set of data) and right (for the other set)

26. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 7 6 1 2 3 5 4

27. Chapter 2 – Section 2
A dot plot is a graph where a dot is placed over the observation each time it is observed
The following is an example of a dot plot

28. Chapter 2 – Section 2 Learning objectives
Organize discrete data in tables
Construct histograms of discrete data
Organize continuous data in tables
Construct histograms of continuous data
Draw stem-and-leaf plots
Draw dot plots
Identify the shape of a distribution 7 6 1 2 3 5 4

29. Chapter 2 – Section 2
A useful way to describe a variable is by the shape of its distribution
Some common distribution shapes are
Uniform
Bell-shaped (or normal)
Skewed right
Skewed left

30. Chapter 2 – Section 2 A variable has a uniform distribution when
Each of the values tends to occur with the same frequency
The histogram looks flat

31. Chapter 2 – Section 2 A variable has a bell-shaped distribution when
Most of the values fall in the middle
The frequencies tail off to the left and to the right
It is symmetric

32. Chapter 2 – Section 2 A variable has a skewed right distribution when
The distribution is not symmetric
The tail to the right is longer than the tail to the left
The arrow from the middle to the long tail points right
Right

33. Chapter 2 – Section 2 A variable has a skewed left distribution when
The distribution is not symmetric
The tail to the left is longer than the tail to the right
The arrow from the middle to the long tail points left
Left

34. Summary: Chapter 2 – Section 2
Quantitative data can be organized in several ways
Histograms based on data values are good for discrete data
Histograms based on classes (intervals) are good for continuous data
The shape of a distribution describes a variable … histograms are useful for identifying the shapes