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Chapter 2: Graphical Descriptions of Data Chapter 2: Graphical Descriptions of Data In chapter 1, you were introduced to the concept of a population, which again is the set of all individuals of interest. Remember, in most cases you can’t collect data from the entire population, so you have to take a sample. Thus, you collect data either through a sample or a census. Now you have a large number of data values. What can you do with them? No one likes to look at just a set of numbers. One thing is to organize the data into a table or graph. Ultimately though, you want to be able to use that graph to interpret the data, to describe the distribution of the data set, and to explore different characteristics of the data. The characteristics that will be discussed in this chapter and the next chapter are: 1. Center: middle of the data set, also known as the average. 2. Variation: how much the data varies. 3. Distribution: shape of the data (symmetric, uniform, or skewed). 4. Outliers: data values that are far from the majority of the data. This chapter will focus mostly on using the graphs to understand aspects of the data, and not as much on how to create the graphs. There is technology that will create most of the graphs, though it is important for you to understand the basics of how to create them. Section 2.1: Qualitative Data Remember, qualitative data are words describing a characteristic of the individual (including numbers that don’t count or measure anything about the individual). There are several different graphs that are used for qualitative data. Qualitative data can first be organized in a frequency or relative frequency table. Frequency table – Relative frequency table – Relative frequency tables are useful when comparing data sets where the sample sizes are not the same. Example #2.1.1: Creating a Frequency Table for Qualitative Data Suppose you have the following data for which type of car students at a campus drive. Ford, Chevy, Honda, Toyota, Toyota, Nissan, Kia, Nissan, Chevy, Toyota, Honda, Chevy, Toyota, Nissan, Ford, Toyota, Nissan, Mercedes, Chevy, Ford, Nissan, Toyota, Nissan, Ford, Chevy, Toyota, Nissan, Honda, Porsche, Hyundai, Chevy, Chevy, Honda, Toyota, Chevy, Ford, Nissan, Toyota, Chevy, Honda, Chevy, Saturn, Toyota, Chevy, Chevy, Nissan, Honda, Toyota, Toyota, Nissan 22

Chapter 2: Graphical Descriptions of Data First identify the individual, variable and type of variable. Individual: Variable: Type of variable: A listing of data is too hard to look at and analyze, so you need to summarize it. First you need to decide the categories. In this case it is relatively easy; just use the car type. However, there are several cars that only have one car in the list. In that case it is easier to make a category called “other” for the ones with low values. Now just count how many of each type of cars there are. For example, there are 5 Fords, 12 Chevys, and 6 Hondas. This can be put in a frequency distribution: Table #2.1.1: Frequency Table for Type of Car Data Category Frequency Ford Chevy Honda Toyota Nissan Other Total 50 The total of the frequency column should be the number of observations in the data. Typically, the counts are not what are reported. Instead, the relative frequencies are used. This is just the frequency divided by the total. As an example for the Ford category: This can be written as a decimal, fraction, or percent. You now have a relative frequency distribution: Table #2.1.2: Relative Frequency Table for Type of Car Data Category Frequency Relative Frequency Ford 5 Chevy 12 Honda 6 Toyota 12 Nissan 10 Other 5 Total 50 1.00 The relative frequency column should add up to 1.00. It might be off a little due to rounding errors on certain problems. 23

Chapter 2: Graphical Descriptions of Data TECHNOLOGY: ENTERING OR UPLOADING DATA INTO STATCRUNCH Entering your own data that you do not have in a file:  Go to Statcrunch.com and login.  Click “Open StatCrunch”. A spreadsheet will open where you can rename the columns using the variable names for your data. You can then enter the raw data into the columns. Entering data from a file:  For the examples and homework problems in this book, you have a file in Blackboard called “Chapter 2 Data”. Save that file to your desktop.  Go to Statcrunch.com and login.  Click “MyStatCrunch”. Under “My Data”, click “Select a file from my computer”. Then choose the file you just saved to your desktop.  Then scroll down and click “Load file” and you will see the data automatically load into the columns of the spreadsheet.  This file is automatically saved under “My Data”. So the next time you login to StatCrunch, you can click “MyStatCrunch” and then click “My Data” and this file will be in the list to choose. TECHNOLOGY: FREQUENCY AND RELATIVE FREQUENCY TABLES IN STATCRUNCH  Enter the data into a column in the spreadsheet (see earlier instructions on entering a list of data)  Click Stat, Tables, Frequency  In the popup window that opens, choose the variable name from “Select Columns”  Under “Statistics” Frequency and Relative Frequency are already chosen so you do not need to click anything there.  Under “Order by” you can choose “Values ascending” to put the categories in ABC order in the table, or “Count ascending” to put the categories in order by frequency, or “Worksheet” to put the categories in order of appearance in the column of data.  Under “”Other*” if percent less than” you can enter a number like 10 to put all categories with less than 10% into a combined category called “Other*”  Then click “Compute!” 24

Chapter 2: Graphical Descriptions of Data If you follow the StatCrunch directions above for the list of data called “Car Data” from the “Chapter02DataFile”, you will get the following: Now that you have the frequency and relative frequency table, it would be good to display this data using a graph. The most common graphs for qualitative data are bar charts and pie charts. Bar chart (or graph) – consist of the frequencies on one axis and the categories on the other axis. Then you draw rectangles for each category with a height (if frequency is on the vertical axis) or length (if frequency is on the horizontal axis) that is equal to the frequency. All of the rectangles should be the same width, and there should be equally width gaps between each bar. Pie chart (or graph) – consists of a circle divided into sectors (pie shapes) that are proportional to the size of the frequency or relative frequency of each category. All you have to do to find the angle is to multiply the relative frequency by 360 degrees. Remember that 180 degrees is half of a circle and 90 degrees is a quarter of a circle. We will be using technology to make these, so you will not need to do these calculations. Example #2.1.2: Drawing a Bar Chart Draw a bar chart of the data in example #2.1.1. Table #2.1.2: Frequency Table for Type of Car Data Category Frequency Relative Frequency Ford 5 0.10 Chevy 12 0.24 Honda 6 0.12 Toyota 12 0.24 Nissan 10 0.20 Other 5 0.10 Total 50 1.00 Put the frequency on the vertical axis and the category on the horizontal axis. Then just draw a box above each category whose height is the frequency. 25

Chapter 2: Graphical Descriptions of Data TECHNOLOGY: BAR CHARTS (BAR GRAPHS) FROM RAW DATA Using StatCrunch :  Enter the data into a column in the spreadsheet (see earlier instructions on entering a list of data)  Click Graph, Bar Plot, With Data  In the popup window that opens choose the variable name from “Select Columns” and under “Type” choose frequency or relative frequency depending on what you have been asked for.  Under “Order by” you can choose “Values ascending” to put the categories in ABC order on the axis, or “Count ascending” to put the bars in order by height, or “Worksheet” to put the categories in order of appearance in the column of data.  Under “”Other*” if percent less than” you can enter a number like 10  Under “Display” check next to “Value above bar”  Under “Graph properties” you can give your graph a title.  Then click “Compute!” If you follow the StatCrunch directions above for the list of raw data called “Car Data” in the “Chapter02DataFile” you will get the following frequency bar chart (value ascending and any category with less than 10% was put into an “Other*” category): Graph #2.1.1: Bar Chart for Type of Car Data Notice from the graph, you can see that Toyota and Chevy are the more popular car, with Nissan not far behind. Ford seems to be the type of car that you can tell was the least liked, though the cars in the other category would be liked less than a Ford. 26

Chapter 2: Graphical Descriptions of Data Some key features of a bar graph:  Equal spacing on each axis.  Bars are the same width.  There should be labels on each axis and a title for the graph.  There should be an equal scaling on the frequency/relative frequency axis and the categories should be listed on the category axis.  The bars don’t touch. You can also draw a bar graph using relative frequency on the vertical axis. This is useful when you want to compare two samples with different sample sizes. The relative frequency graph and the frequency graph should look the same, except for the scaling on the frequency axis. If you follow the StatCrunch directions above for the list of data called “Car Data” in the “Chapter02DataFile” you will get the following relative frequency bar chart (value ascending and any category with less than 10% was put into an “Other*” category): Graph #2.1.2: Relative Frequency Bar Chart for Type of Car Data If instead you had chosen “Count descending” the bar plot would look as follows: 27

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