Organizing Quantitative Data: Tables, Histograms, Stem-and-Leaf Plots

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

2. 7 6 1 2 3 5 4 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

3. 1 2 3 4 6 5 7 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

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. 1 2 3 4 6 5 7 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

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. 1 2 3 4 6 5 7 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

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 lowerclasslimit • 39 is the upperclasslimit

14. Chapter 2 – Section 2 • The classwidth 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 • The classwidth 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

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 • 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-endedclass because it has no upper limit • 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-endedclass because it has no upper limit • Classes with no lower limits are also called open-ended classes

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

17. Chapter 2 – Section 2 • 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 • 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) • 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 • 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

18. 1 2 3 6 4 7 5 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

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. 1 2 3 6 4 7 5 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

21. Chapter 2 – Section 2 • A stem-and-leafplot is a different way to represent data that is similar to a histogram • A stem-and-leafplot 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”

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 • Read the stem first • Attach the leaf as the last digit of the stem • The result is the original data value • 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 • 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)

24. Chapter 2 – Section 2 • 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 • 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 • 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 • 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

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 10-14 in one stem and 15-19 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 10-14 in one stem and 15-19 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) • 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 10-14 in one stem and 15-19 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)

26. 1 2 3 4 5 7 6 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

27. Chapter 2 – Section 2 • A dotplot 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. 1 2 3 4 5 6 7 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

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 skewedright 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 skewedleft 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