Chapter 3

1. Organizing, Displaying, and Interpreting Data Chapter 3

2. Organizing, Displaying, and Interpreting Data Section 3aFrequency Distributions HAWKES LEARNING SYSTEMS math courseware specialists Objectives: • Learn how to construct a frequency distribution. • Know the characteristics of a frequency distribution.

3. Organizing, Displaying, and Interpreting Data Section 3.1Frequency Distributions HAWKES LEARNING SYSTEMS math courseware specialists Definition: • A frequency distribution is a summary technique that organizes data into classes and provides in tabular form a list of the classes along with the number of observations in each class. • The two steps in constructing frequency distributions are: • Choosing the classifications, and • Counting the number in each class. • Choosing the type of classification depends on whether the data is qualitative (nominal or ordinal) or quantitative (interval or ratio).

4. Organizing, Displaying, and Interpreting Data Section 3bGraphical Displays of Data: Pie Charts and Bar Graphs HAWKES LEARNING SYSTEMS math courseware specialists Objectives: • Create the basic types of pie charts and bar graphs. • Interpret data given in pie charts and bar graphs.

5. Organizing, Displaying, and Interpreting Data Section 3.2TheValue of Graphs HAWKES LEARNING SYSTEMS math courseware specialists Graphs: • A set of data can be graphically represented in many different ways. • Creating graphical displays requires a certain amount of artistic judgment. • Development of graphical software has made graphing easy. • Types of graphs include: • Bar charts • Pie charts • Line charts • Stem and leaf displays • Histograms

6. Organizing, Displaying, and Interpreting Data Section 3.2TheValue of Graphs HAWKES LEARNING SYSTEMS math courseware specialists Types of Graphs: • Bar charts • Pie charts • Stem and leaf displays • Histograms

7. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically What do you notice about the axis labels? Misleading Graphs: Remember: When you see an axis that doesn’t start at zero, be a bit skeptical of the conclusions the author intends for you to make. The bar graphs below are both plots of the same data set.

8. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists Bar Chart: • The bar chart is a graphical display in which the length of each bar corresponds to the number of observations in a category. • Bar charts are: • used to illustrate a frequency distribution for qualitative data. • valuable as presentation tools. • effective at reinforcing differentials in magnitudes. • comprised of vertical or horizontal bars.

9. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists Bar Chart: As mentioned in the last slide, bar graphs represent qualitative data. Can you tell the categories are qualitative? Specifically, what level of measurement are the categories an example of? Solution: Ordinal

10. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists Conventions of Bar Chart Construction: • Maintain order of categories • Miscellaneous or “other” should be listed at the bottom of horizontal graphs or at the far right in vertical graphs • Effectively choose a scale to allow for desired comparison • Choose visually pleasing bar widths • Do not vary the bar width throughout the chart • Use shading, crosshatching, and color to help present data • The spacing between bars should be set at approximately one-half the width of a bar • Source notes are placed below the chart • Gridlines are often used and increase readability • Label each axis if there is room

11. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists Stacked Bar Charts: • Variation on the standard bar chart • Allows comparison of total quantity as well as the individual quantity of several subcategories. • Example: Grandchildren living with their Grandparents Number of Children

12. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists 3-D Bar Charts: • Below is an example of a 3-D bar chart. • The chart displays the following question from a survey by the Gallup poll: • Do you think women should be permitted to sunbathe on public beaches, or should it be banned?

13. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists Pie Charts: • A pie chart shows us how large each category is in relation to the whole. • Can be used to express frequency distributions. • The circle represents the total “pie” available. • The slices are proportional to the amount in each category. • Each slice of the pie represents the proportion of total observations belonging to the category. • Easy to compare the total in each of the classifications to the total number of observations.

14. Organizing, Displaying, and Interpreting Data Section 3.3Displaying Qualitative Data Graphically HAWKES LEARNING SYSTEMS math courseware specialists Pie Charts: • Most commonly, pie charts are used to display how money is spent. The pie chart below tells an interesting story about how our tax dollars are spent.

15. Organizing, Displaying, and Interpreting Data Section 3cGraphical Displays of Data: Histograms, Polygons, Stem and Leaf Plots HAWKES LEARNING SYSTEMS math courseware specialists Objectives: • Understand how to read and interpret the information shown in line graphs, histograms, frequency polygons, ogives, and stem and leaf plots. • Be able to perform appropriate operations related to the data shown in a line graph, histogram, frequency polygon, ogive, or stem and leaf plot. • Construct histograms, frequency polygons, and ogives from the data given.

16. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data HAWKES LEARNING SYSTEMS math courseware specialists Frequency Distributions: • The purpose of a frequency distribution is to condense a set of data into a meaningful summary form. • Remember there are two steps in the construction of a frequency distribution: • choosing the classifications, and • counting the number in each class.

17. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data HAWKES LEARNING SYSTEMS math courseware specialists Types of Frequency Distributions: • Distributions used to organize data: • Relative Frequency • Cumulative Frequency • Cumulative Relative Frequency

18. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data HAWKES LEARNING SYSTEMS math courseware specialists Selecting the Number of Classes: • The fundamental decision in constructing a frequency distribution is selecting the number of classes. • The number of classes depends on the amount of data available. • Generally fewer than 4 classes compresses the data. • More than 20 classes provides too little summary information. • Once you determine the number of classes, the next step is to specify the class width.

19. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data HAWKES LEARNING SYSTEMS math courseware specialists Determining the Class Width: • Usually, the class widths are equal widths, except for the beginning and ending of intervals. • There is no perfect formula for class width that will work for every data set. However a good starting point for determining class width is:

20. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data Example: Create a frequency distribution with the following heart rate data: Class endpoints with fractional values will make the graph slightly difficult to digest. If possible, try a class width in the range of 8 to 10. An interval width of 10 is used in this example. If there are five classes, determine the class width.

21. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data HAWKES LEARNING SYSTEMS math courseware specialists Relative Frequency: • The relative frequency represents the proportion of the total number of observations in a given class. • Relative frequency: • Allows us to view the number in each category in relation to the total number of observations. • Is a standardizing technique. • Enables us to compare data sets with different numbers of observations.

22. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data Example: Fifty students had their heart rate checked. Find the relative frequency of each interval. .06 .26 .58 .08 .02

23. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data Cumulative Frequency: • The cumulative frequency is the sum of the frequency of a particular class and all preceding classes. Below is a cumulative frequency distribution for the heart rate data. 3 16 45 49 50

24. Organizing, Displaying, and Interpreting Data Section 3.4Constructing Frequency Distributions for Quantitative Data Cumulative Relative Frequency: The cumulative relative frequency is the proportion of observations in a particular class and all preceding classes. Below is a cumulative relative frequency distribution for the heart rate data. 0.32 0.90 0.98 1.00

25. Organizing, Displaying, and Interpreting Data Section 3.5Histograms HAWKES LEARNING SYSTEMS math courseware specialists Histograms: • A histogram is a bar graph of a frequency or relative frequency distribution in which the height of each bar corresponds to the frequency or relative frequency of the class. • A histogram: • is one of the most frequently used statistical tools. • reveals the structure of the data. • is easy to interpret.

26. Organizing, Displaying, and Interpreting Data Section 3.5Histograms Ch 3. Organizing, Displaying, and Interpreting Data 3.5 Histograms Examples of Histograms:

27. Organizing, Displaying, and Interpreting Data Section 3.6The Stem and Leaf Display HAWKES LEARNING SYSTEMS math courseware specialists Stem and Leaf Display: • The stem and leaf display is a hybrid graphical method. • The display is similar to a histogram, but the data remains visible. • Useful in ordering and detecting patterns in the data. • One of the few graphical methods in which raw data is not lost in the construction. • As the name implies there will be a “stem” to which “leaves” will be attached in some pattern.

28. Organizing, Displaying, and Interpreting Data Section 3.6The Stem and Leaf Display Example: Consider the following data: 97, 99, 108, 110, 111. Here we are interested in the variation of the last digit. Make a table first, then construct the stem and leaf display. Notice the leaves are the ones digit and the stems are the tens digit.

29. Organizing, Displaying, and Interpreting Data Section 3.6The Stem and Leaf Display Example: Suppose that now we are interested in the last two digits. Let’s make the table first. Since we are looking for the last two digits we know what to put in the leaf column. Now simply put what's left (if anything) in the stem column. Now construct the stem and leaf display.

30. Organizing, Displaying, and Interpreting Data Section 3.7The Ordered Array HAWKES LEARNING SYSTEMS math courseware specialists Ordered Array: • An ordered array is a listing of all the data in either increasing or decreasing magnitude. • Data listed in increasing order is said to be listed in rank order. • If listed in decreasing order, data is listed in reverse order. • Listing the data in an ordered way can be very helpful. • By ordering the data it enables you to scan the data quickly for the largest and smallest values, for large gaps in data, and for concentrations or clusters in values.

31. Organizing, Displaying, and Interpreting Data Section 3.7The Ordered Array HAWKES LEARNING SYSTEMS math courseware specialists Example: • The personnel records for a clothing department store located in the mall are examined and all the current ages are noted. There are 25 employees, and their ages are all listed below. • Place the ages in rank order. • Solution:

32. Organizing, Displaying, and Interpreting Data Section 3.8Dot Plots HAWKES LEARNING SYSTEMS math courseware specialists Dot Plot: • A dot plot is a graph where each of the data values is plotted as a point on the horizontal axis. • If there is a multitude of entries of the same data value, they are plotted one above the other.

33. Organizing, Displaying, and Interpreting Data Section 3.9Plotting Time Series Data HAWKES LEARNING SYSTEMS math courseware specialists Time Series Plot: • A time series plot graphs data using time as the horizontal axis. • Time series data can be represented in many different ways including bar graphs, line graphs, or 3-D line graphs.