# Chapter 2: Graphical Descriptions of Data

Chapter 2: Graphical Descriptions of Data In chapter 1, you were introduced to the concepts of population, which again is a collection of all the measurements from the individuals of interest. Remember, in most cases you can’t collect 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. Qualitative data: analysis of the data 5. Outliers: data values that are far from the majority of the data. 6. Time: changing characteristics of the data over time.

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. There are several different graphs that are used for qualitative data. These graphs include bar graphs, Pareto charts, and pie charts. Pie charts and bar graphs are the most common ways of displaying qualitative data. A spreadsheet program like Excel can make both of them. The first step for either graph is to make a frequency or relative frequency table. A frequency table is a summary of the data with counts of how often a data value (or category) occurs. Example #2.1.1: Creating a Frequency Table

Suppose you have the following data for which type of car students at a college 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

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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 5 Chevy 12 Honda 6 Toyota 12 Nissan 10 Other 5 Total 50

The total of the frequency column should be the number of observations in the data.

Since raw numbers are not as useful to tell other people it is better to create a third column that gives the relative frequency of each category. This is just the frequency divided by the total. As an example for Ford category:

relative frequency = 5

50 = 0.10

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 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

The relative frequency column should add up to 1.00. It might be off a little due to rounding errors.

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Now that you have the frequency and relative frequency table, it would be good to display this data using a graph. There are several different types of graphs that can be used: bar chart, pie chart, and Pareto charts. Bar graphs or charts 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. Example #2.1.2: Drawing a Bar Graph Draw a bar graph 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. (All graphs are drawn using Microsoft Excel.)

Graph #2.1.1: Bar Graph for Type of Car Data

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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.

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 a scaling on the 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. Graph #2.1.2: Relative Frequency Bar Graph for Type of Car Data

Another type of graph for qualitative data is a pie chart. A pie chart is where you have a circle and you divide pieces of the circle into pie shapes that are proportional to the size of the relative frequency. There are 360 degrees in a full circle. Relative frequency is just the percentage as a decimal. All you have to do to find the angle by multiplying the relative frequency by 360 degrees. Remember that 180 degrees is half a circle and 90 degrees is a quarter of a circle.

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Example #2.1.3: Drawing a Pie Chart Draw a pie chart of the data in example #2.1.1. First you need the relative frequencies. 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

Then you multiply each relative frequency by 360° to obtain the angle measure for each category.

Table #2.1.3: Pie Chart Angles for Type of Car Data

Category Relative

Frequency

Angle (in degrees

(°)) Ford 0.10 36.0 Chevy 0.24 86.4 Honda 0.12 43.2 Toyota 0.24 86.4 Nissan 0.20 72.0 Other 0.10 36.0 Total 1.00 360.0

Now draw the pie chart.

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Graph #2.1.3: Pie Chart for Type of Car Data

As you can see from the graph, Toyota and Chevy are more popular, while the cars in the other category are liked the least. Of the cars that you can determine from the graph, Ford is liked less than the others.

Pie charts are useful for comparing sizes of categories. Bar charts show similar information. It really doesn’t matter which one you use. It really is a personal preference and also what information you are trying to address. However, pie charts are best when you only have a few categories and the data can be expressed as a percentage. The data doesn’t have to be percentages to draw the pie chart, but if a data value can fit into multiple categories, you cannot use a pie chart. As an example, if you asking people about what their favorite national park is, and you say to pick the top three choices, then the total number of answers can add up to more than 100% of the people involved. So you cannot use a pie chart to display the favorite national park. A third type of qualitative data graph is a Pareto chart, which is just a bar chart with the bars sorted with the highest frequencies on the left. Here is the Pareto chart for the data in Example #2.1.1.

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Graph #2.1.4: Pareto Chart for Type of Car Data

The advantage of Pareto charts is that you can visually see the more popular answer to the least popular. This is especially useful in business applications, where you want to know what services your customers like the most, what processes result in more injuries, which issues employees find more important, and other type of questions like these. There are many other types of graphs that can be used on qualitative data. There are spreadsheet software packages that will create most of them, and it is better to look at them to see what can be done. It depends on your data as to which may be useful. The next example illustrates one of these types known as a multiple bar graph. Example #2.1.4: Multiple Bar Graph

In the Wii Fit game, you can do four different types if exercises: yoga, strength, aerobic, and balance. The Wii system keeps track of how many minutes you spend on each of the exercises everyday. The following graph is the data for Dylan over one week time period. Discuss any indication you can infer from the graph.

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Graph #2.1.5: Multiple Bar Chart for Wii Fit Data

Solution: It appears that Dylan spends more time on balance exercises than on any other exercises on any given day. He seems to spend less time on strength exercises on a given day. There are several days when the amount of exercise in the different categories is almost equal.

The usefulness of a multiple bar graph is the ability to compare several different categories over another variable, in example #2.1.4 the variable would be time. This allows a person to interpret the data with a little more ease. Section 2.1: Homework 1.) Eyeglassomatic manufactures eyeglasses for different retailers. The number of

lenses for different activities is in table #2.1.4. Table #2.1.4: Data for Eyeglassomatic Activity Grind Multicoat Assemble Make

Unknown

Number of lenses

18872 12105 4333 25880 26991 1508

Grind means that they ground the lenses and put them in frames, multicoat means that they put tinting or scratch resistance coatings on lenses and then put them in frames, assemble means that they receive frames and lenses from other sources and put them together, make frames means that they make the frames and put lenses in from other sources, receive finished means that they received glasses from other source, and unknown means they do not know where the lenses came from. Make a bar chart and a pie chart of this data. State any findings you can see from the graphs.

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2.) To analyze how Arizona workers ages 16 or older travel to work the percentage of workers using carpool, private vehicle (alone), and public transportation was collected. Create a bar chart and pie chart of the data in table #2.1.5. State any findings you can see from the graphs.

Table #2.1.5: Data of Travel Mode for Arizona Workers

Transportation type Percentage Carpool 11.6% Private Vehicle (Alone) 75.8% Public Transportation 2.0% Other 10.6%

3.) The number of deaths in the US due to carbon monoxide (CO) poisoning from

generators from the years 1999 to 2011 are in table #2.1.6 (Hinatov, 2012). Create a bar chart and pie chart of this data. State any findings you see from the graphs.

Table #2.1.6: Data of Number of Deaths Due to CO Poisoning Region Number of deaths from CO

while using a generator Urban Core 401 Sub-Urban 97 Large Rural 86 Small Rural/Isolated 111

4.) In Connecticut households use gas, fuel oil, or electricity as a heating source.

Table #2.1.7 shows the percentage of households that use one of these as their principle heating sources (“Electricity usage,” 2013), (“Fuel oil usage,” 2013), (“Gas usage,” 2013). Create a bar chart and pie chart of this data. State any findings you see from the graphs.

Table #2.1.7: Data of Household Heating Sources

Heating Source Percentage Electricity 15.3% Fuel Oil 46.3% Gas 35.6% Other 2.8%

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5.) Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made during the time period of January 1 to March 31. Table #2.1.8 gives the defect and the number of defects. Create a Pareto chart of the data and then describe what this tells you about what causes the most defects.

Table #2.1.8: Data of Defect Type Defect type Number of defects Scratch 5865 Right shaped – small 4613 Flaked 1992 Wrong axis 1838 Chamfer wrong 1596 Crazing, cracks 1546 Wrong shape 1485 Wrong PD 1398 Spots and bubbles 1371 Wrong height 1130 Right shape – big 1105 Lost in lab 976 Spots/bubble – intern 976

6.) People in Bangladesh were asked to state what type of birth control method they

use. The percentages are given in table #2.1.9 (“Contraceptive use,” 2013). Create a Pareto chart of the data and then state any findings you can from the graph.

Table #2.1.9: Data of Birth Control Type Method Percentage Condom 4.50% Pill 28.50% Periodic Abstinence 4.90% Injection 7.00% Female Sterilization 5.00% IUD 0.90% Male Sterilization 0.70% Withdrawal 2.90% Other Modern Methods 0.70% Other Traditional Methods 0.60%

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7.) The percentages of people who use certain contraceptives in Central American countries are displayed in graph #2.1.6 (“Contraceptive use,” 2013). State any findings you can from the graph. Graph #2.1.6: Multiple Bar Chart for Contraceptive Types

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Section 2.2: Quantitative Data The graph for quantitative data looks similar to a bar graph, except there are some major differences. First, in a bar graph the categories can be put in any order on the horizontal axis. There is no set order for these data values. You can’t say how the data is distributed based on the shape, since the shape can change just by putting the categories in different orders. With quantitative data, the data are in specific orders, since you are dealing with numbers. With quantitative data, you can talk about a distribution, since the shape only changes a little bit depending on how many categories you set up. This is called a frequency distribution. This leads to the second difference from bar graphs. In a bar graph, the categories that you made in the frequency table were determined by you. In quantitative data, the categories are numerical categories, and the numbers are determined by how many categories (or what are called classes) you choose. If two people have the same number of categories, then they will have the same frequency distribution. Whereas in qualitative data, there can be many different categories depending on the point of view of the author. The third difference is that the categories touch with quantitative data, and there will be no gaps in the graph. The reason that bar graphs have gaps is to show that the categories do not continue on, like they do in quantitative data. Since the graph for quantitative data is different from qualitative data, it is given a new name. The name of the graph is a histogram. To create a histogram, you must first create the frequency distribution. The idea of a frequency distribution is to take the interval that the data spans and divide it up into equal subintervals called classes. Summary of the steps involved in making a frequency distribution:

1. Find the range = largest value – smallest value 2. Pick the number of classes to use. Usually the number of classes is between

five and twenty. Five classes are used if there are a small number of data points and twenty classes if there are a large number of data points (over 1000 data points). (Note: categories will now be called classes from now on.)

3. Class width = range

# classes . Always round up to the next integer (even if the

answer is already a whole number go to the next integer). If you don’t do this, your last class will not contain your largest data value, and you would have to add another class just for it. If you round up, then your largest data value will fall in the last class, and there are no issues.

4. Create the classes. Each class has limits that determine which values fall in each class. To find the class limits, set the smallest value as the lower class limit for the first class. Then add the class width to the lower class limit to get the next lower class limit. Repeat until you get all the classes. The upper class limit for a class is one less than the lower limit for the next class.

5. In order for the classes to actually touch, then one class needs to start where the previous one ends. This is known as the class boundary. To find the class

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boundaries, subtract 0.5 from the lower class limit and add 0.5 to the upper class limit.

6. Sometimes it is useful to find the class midpoint. The process is

Midpoint = lower limit + upper limit

2

7. To figure out the number of data points that fall in each class, go through each data value and see which class boundaries it is between. Utilizing tally marks may be helpful in counting the data values. The frequency for a class is the number of data values that fall in the class.

Note: the above description is for data values that are whole numbers. If you data value has decimal places, then your class width should be rounded up to the nearest value with the same number of decimal places as the original data. In addition, your class boundaries should have one more decimal place than the original data. As an example, if your data have one decimal place, then the class width would have one decimal place, and the class boundaries are formed by adding and subtracting 0.05 from each class limit. Example #2.2.1: Creating a Frequency Table

Table #2.21 contains the amount of rent paid every month for 24 students from a statistics course. Make a relative frequency distribution using 7 classes.

Table #2.2.1: Data of Monthly Rent

1500 1350 350 1200 850 900 1500 1150 1500 900 1400 1100 1250 600 610 960 890 1325 900 800 2550 495 1200 690

Solution: 1) Find the range:

largest value − smallest value = 2550 − 350 = 2200

2) Pick the number of classes:

The directions say to use 7 classes.

3) Find the class width:

width = range 7

= 2200 7

≈ 314.286

Round up to 315.

Always round up to the next integer even if the width is already an integer.

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4) Find the class limits:

Start at the smallest value. This is the lower class limit for the first class. Add the width to get the lower limit of the next class. Keep adding the width to get all the lower limits. 350 + 315 = 665, 665 + 315 = 980, 980 + 315 = 1295, …

The upper limit is one less than the next lower limit: so for the first class the upper class limit would be 665 −1= 664 .

When you have all 7 classes, make sure the last number, in this case the 2550, is at least as large as the largest value in the data. If not, you made a mistake somewhere.

5) Find the class boundaries:

Subtract 0.5 from the lower class limit to get the class boundaries. Add 0.5 to the upper class limit for the last class’s boundary.

350 − 0.5 = 349.5, 665 − 0.5 = 664.5, 980 − 0.5 = 979.5, 1295 − 0.5 = 1294.5, …

Every value in the data should fall into exactly one of the classes. No data values should fall right on the boundary of two classes.

6) Find the class midpoints:

midpoint = lower limit + upper limit 2

350 + 664 2

= 507, 665 + 979 2

= 822, …

7) Tally and find the frequency of the data:

Go through the data and put a tally mark in the appropriate class for each piece of data by looking to see which class boundaries the data value is between. Fill in the frequency by changing each of the tallies into a number.

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Table #2.2.2: Frequency Distribution for Monthly Rent

Class Limits

Class Boundaries

Class Midpoint

Tally

Frequency

350 – 664 349.5 – 664.5 507 |||| 4 665 – 979 664.5 – 979.5 822 |||| ||| 8

980 – 1294 979.5 – 1294.5 1137 |||| 5 1295 – 1609 1294.5 – 1609.5 1452 |||| | 6 1610 – 1924 1609.5 – 1924.5 1767 0 1925 – 2239 1924.5 – 2239.5 2082 0 2240 – 2554 2239.5 – 2554.5 2397 | 1

Make sure the total of the frequencies is the same as the number of data points.

It is difficult to determine the basic shape of the distribution by looking at the frequency distribution. It would be easier to look at a graph. The graph of a frequency distribution for quantitative data is called a frequency histogram or just histogram for short. Histogram: a graph of the frequencies on the vertical axis and the class boundaries on the horizontal axis. Rectangles where the height is the frequency and the width is the class width are draw for each class. Example #2.2.2: Drawing a Histogram Draw a histogram for the distribution from example #2.2.1.

Solution: The class boundaries are plotted on the horizontal axis and the frequencies are plotted on the vertical axis. You can plot the midpoints of the classes instead of the class boundaries. Graph #2.2.1 was created using the midpoints because it was easier to do with the software that created the graph.

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Graph #2.2.1: Histogram for Monthly Rent

Notice the graph has the axes labeled, the tick marks are labeled on each axis, and there is a title. Reviewing the graph you can see that most of the students pay around \$822 per month for rent, with \$1452 being the other common value. You can see from the graph, that most students pay between \$822 and \$1452 per month for rent. Of course, that is just the midpoint of each of the classes, but you can say on average the students in each class pay around the midpoint. There is a large gap between the \$1452 class and the highest data value. This seems to say that one student is paying a great deal more than everyone else. This value could be considered an outlier. An outlier is a data value that is far from the rest of the values. It may be an unusual value or a mistake. It is a data value that should be investigated. In this case, the student lives in a very expensive part of town, thus the value is not a mistake, and is just very unusual. There are other aspects that can be discussed, but first some other concepts need to be introduced.

Frequencies are helpful, but understanding the relative size each class is to the total is also useful. To find this you can divide the frequency by the total to create a relative frequency. If you have the relative frequencies for all of the classes, then you have a relative frequency distribution. Relative Frequency Distribution A variation on a frequency distribution is a relative frequency distribution. Instead of giving the frequencies for each class, the relative frequencies are calculated.

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Relative frequency = frequency

# of data points

This gives you percentages of data that fall in each class. Example #2.2.3: Creating a Relative Frequency Table

Find the relative frequency for the grade data. Solution: From example #2.2.1, the frequency distribution is reproduced in table #2.2.2.

Table #2.2.2: Frequency Distribution for Monthly Rent

Class Limits

Class Boundaries

Class Midpoint

Frequency

350 – 664 349.5 – 664.5 507 4 665 – 979 664.5 – 979.5 822 8

980 – 1294 979.5 – 1294.5 1137 5 1295 – 1609 1294.5 – 1609.5 1452 6 1610 – 1924 1609.5 – 1924.5 1767 0 1925 – 2239 1924.5 – 2239.5 2082 0 2240 – 2554 2239.5 – 2554.5 2397 1

Divide each frequency by the number of data points.

4 24

= 0.17, 8 24

= 0.33, 5 24

= 0.21, …

Table #2.2.3: Relative Frequency Distribution for Monthly Rent

Class Limits

Class Boundaries

Class Midpoint

Frequency

Relative Frequency

350 – 664 349.5 – 664.5 507 4 0.17 665 – 979 664.5 – 979.5 822 8 0.33

980 – 1294 979.5 – 1294.5 1137 5 0.21 1295 – 1609 1294.5 – 1609.5 1452 6 0.25 1610 – 1924 1609.5 – 1924.5 1767 0 0 1925 – 2239 1924.5 – 2239.5 2082 0 0 2240 – 2554 2239.5 – 2554.5 2397 1 0.04

Total 24 1

The relative frequencies should add up to 1 or 100%. (This might be off a little due to rounding errors.)

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The graph of the relative frequency is known as a relative frequency histogram. It looks identical to the frequency histogram, but the vertical axis is relative frequency instead of just frequencies. Example #2.2.4: Drawing a Relative Frequency Histogram

Draw a relative frequency histogram for the grade distribution from example #2.2.1.

Solution: The class boundaries are plotted on the horizontal axis and the relative frequencies are plotted on the vertical axis.

Graph #2.2.2: Relative Frequency Histogram for Monthly Rent

Notice the shape is the same as the frequency distribution.

Another useful piece of information is how many data points fall below a particular class boundary. As an example, a teacher may want to know how many students received below an 80%, a doctor may want to know how many adults have cholesterol below 160, or a manager may want to know how many stores gross less than \$2000 per day. This is known as a cumulative frequency. If you want to know what percent of the data falls below a certain class boundary, then this would be a cumulative relative frequency. For cumulative frequencies you are finding how many data values fall below the upper class limit.

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To create a cumulative frequency distribution, count the number of data points that are below the upper class boundary, starting with the first class and working up to the top class. The last upper class boundary should have all of the data points below it. Also include the number of data points below the lowest class boundary, which is zero. Example #2.2.5: Creating a Cumulative Frequency Distribution Create a cumulative frequency distribution for the data in example #2.2.1.

Solution: The frequency distribution for the data is in table #2.2.2. Table #2.2.2: Frequency Distribution for Monthly Rent

Class Limits

Class Boundaries

Class Midpoint

Frequency

350 – 664 349.5 – 664.5 507 4 665 – 979 664.5 – 979.5 822 8

980 – 1294 979.5 – 1294.5 1137 5 1295 – 1609 1294.5 – 1609.5 1452 6 1610 – 1924 1609.5 – 1924.5 1767 0 1925 – 2239 1924.5 – 2239.5 2082 0 2240 – 2554 2239.5 – 2554.5 2397 1

Now ask yourself how many data points fall below each class boundary. Below 349.5, there are 0 data points. Below 664.5 there are 4 data points, below 979.5, there are 4 + 8 = 12 data points, below 1294.5 there are 4 + 8 + 5 = 17 data points, and continue this process until you reach the upper class boundary. This is summarized in the table below.

Table #2.2.4: Cumulative Distribution for Monthly Rent

Class Limits

Class Boundaries

Class Midpoint

Frequency

Cumulative Frequency

350 – 664 349.5 – 664.5 507 4 4 665 – 979 664.5 – 979.5 822 8 12

980 – 1294 979.5 – 1294.5 1137 5 17 1295 – 1609 1294.5 – 1609.5 1452 6 23 1610 – 1924 1609.5 – 1924.5 1767 0 23 1925 – 2239 1924.5 – 2239.5 2082 0 23 2240 – 2554 2239.5 – 2554.5 2397 1 24

Again, it is hard to look at the data the way it is. A graph would be useful. The graph for cumulative frequency is called an ogive (o-jive). To create an ogive, first create a scale

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on both the horizontal and vertical axes that will fit the data. Then plot the points of the class upper class boundary versus the cumulative frequency. Make sure you include the point with the lowest class boundary and the 0 cumulative frequency. Then just connect the dots. Example #2.2.6: Drawing an Ogive Draw an ogive for the data in example #2.2.1.

Solution: Graph #2.2.3: Ogive for Monthly Rent

The usefulness of a ogive is to allow the reader to find out how many students pay less than a certain value, and also what amount of monthly rent is paid by a certain number of students. As an example, suppose you want to know how many students pay less than \$1500 a month in rent, then you can go up from the \$1500 until you hit the graph and then you go over to the cumulative frequency axes to see what value corresponds to this value. It appears that around 20 students pay less than \$1500. (See graph #2.2.4.)

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Graph #2.2.4: Ogive for Monthly Rent with Example

Also, if you want to know the amount that 15 students pay less than, then you start at 15 on the vertical axis and then go over to the graph and down to the horizontal axis where the line intersects the graph. You can see that 15 students pay less than about \$1200 a month. (See graph #2.2.5.) Graph #2.2.5: Ogive for Monthly Rent with Example

If you graph the cumulative relative frequency then you can find out what percentage is below a certain number instead of just the number of people below a certain value.

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Shapes of the distribution: When you look at a distribution, look at the basic shape. There are some basic shapes that are seen in histograms. Realize though that some distributions have no shape. The common shapes are symmetric, skewed, and uniform. Another interest is how many peaks a graph may have. This is known as modal.

Symmetric means that you can fold the graph in half down the middle and the two sides will line up. You can think of the two sides as being mirror images of each other. Skewed means one “tail” of the graph is longer than the other. The graph is skewed in the direction of the longer tail (backwards from what you would expect). A uniform graph has all the bars the same height. Modal refers to the number of peaks. Unimodal has one peak and bimodal has two peaks. Usually if a graph has more than two peaks, the modal information is not longer of interest. Other important features to consider are gaps between bars, a repetitive pattern, how spread out is the data, and where the center of the graph is. Examples of graphs:

This graph is roughly symmetric and unimodal:

Graph #2.2.6: Symmetric, Unimodal Graph

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This graph is symmetric and bimodal:

Graph #2.2.7: Symmetric, Bimodal Graph

This graph is skewed to the right:

Graph #2.2.8: Skewed Right Graph

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This graph is skewed to the left and has a gap:

Graph #2.2.9: Skewed Left Graph

This graph is uniform since all the bars are the same height:

Graph #2.2.10: Uniform Graph

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Example #2.2.7: Creating a Frequency Distribution, Histogram, and Ogive The following data represents the percent change in tuition levels at public, four- year colleges (inflation adjusted) from 2008 to 2013 (Weissmann, 2013). Create a frequency distribution, histogram, and ogive for the data.

Table #2.2.5: Data of Tuition Levels at Public, Four-Year Colleges

19.5% 40.8% 57.0% 15.1% 17.4% 5.2% 13.0% 15.6% 51.5% 15.6% 14.5% 22.4% 19.5% 31.3% 21.7% 27.0% 13.1% 26.8% 24.3% 38.0% 21.1% 9.3% 46.7% 14.5% 78.4% 67.3% 21.1% 22.4% 5.3% 17.3% 17.5% 36.6% 72.0% 63.2% 15.1% 2.2% 17.5% 36.7% 2.8% 16.2% 20.5% 17.8% 30.1% 63.6% 17.8% 23.2% 25.3% 21.4% 28.5% 9.4%

Solution:

1) Find the range:

largest value − smallest value = 78.4%− 2.2% = 76.2%

2) Pick the number of classes:

Since there are 50 data points, then around 6 to 8 classes should be used. Let’s use 8.

3) Find the class width:

width = range 8

= 76.2% 8

≈ 9.525%

Since the data has one decimal place, then the class width should round to one decimal place. Make sure you round up.

width = 9.6%

4) Find the class limits:

2.2%+ 9.6% = 11.8%, 11.8%+ 9.6% = 21.4%, 21.4%+ 9.6% = 31.0%,…

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5) Find the class boundaries:

Since the data has one decimal place, the class boundaries should have two decimal places, so subtract 0.05 from the lower class limit to get the class boundaries. Add 0.05 to the upper class limit for the last class’s boundary.

2.2 − 0.05 = 2.15%, 11.8 − 0.05 = 11.75%, 21.4 − 0.05 = 21.35%, … Every value in the data should fall into exactly one of the classes. No data values should fall right on the boundary of two classes.

6) Find the class midpoints:

midpoint = lower limt + upper limit 2

2.2 +11.7 2

= 6.95%, 11.8 + 21.3 2

= 16.55%, …

7) Tally and find the frequency of the data:

Table #2.2.6: Frequency Distribution for Tuition Levels at Public, Four-Year Colleges

Class Limits

Class Boundaries

Class Midpoint

Tally

Frequency

Relative Frequency

Cumulative Frequency

2.2 – 11.7 2.15 – 11.75 6.95 |||| | 6 0.12 6

11.8 – 21.3 11.75 – 21.35 16.55 |||| |||| |||| ||||

20 0.40 26

21.4 – 30.9 21.35 – 30.95 26.15 |||| |||| | 11 0.22 37 31.0 – 40.5 30.95 – 40.55 35.75 |||| 4 0.08 41 40.6 – 50.1 40.55 – 50.15 45.35 || 2 0.04 43 50.2 – 59.7 50.15 – 59.75 54.95 || 2 0.04 45 59.8 – 69.3 59.75 – 69.35 64.55 ||| 3 0.06 48 69.4 – 78.9 69.35 – 78.95 74.15 || 2 0.04 50

Make sure the total of the frequencies is the same as the number of data points.

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Graph #2.2.11: Histogram for Tuition Levels at Public, Four-Year Colleges

This graph is skewed right, with no gaps. This says that most percent increases in tuition were around 16.55%, with very few states having a percent increase greater than 45.35%.

Graph #2.2.11: Ogive for Tuition Levels at Public, Four-Year Colleges

Looking at the ogive, you can see that 30 states had a percent change in tuition levels of about 25% or less.

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There are occasions where the class limits in the frequency distribution are predetermined. Example #2.2.8 demonstrates this situation. Example #2.2.8: Creating a Frequency Distribution and Histogram

The following are the percentage grades of 25 students from a statistics course. Make a frequency distribution and histogram.

Table #2.2.7: Data of Test Grades

62 87 81 69 87 62 45 95 76 76 62 71 65 67 72 80 40 77 87 58 84 73 93 64 89

Solution:

Since this data is percent grades, it makes more sense to make the classes in multiples of 10, since grades are usually 90 to 100%, 80 to 90%, and so forth. It is easier to not use the class boundaries, but instead use the class limits and think of the upper class limit being up to but not including the next classes lower limit. As an example the class 80 – 90 means a grade of 80% up to but not including a 90%. A student with an 89.9% would be in the 80-90 class. Table #2.2.8: Frequency Distribution for Test Grades

Class Limit Class

Midpoint Tally Frequency 40 – 50 45 || 2 50 – 60 55 | 1 60 – 70 65 |||| || 7 70 – 80 75 |||| | 6 80 – 90 85 |||| || 7 90 – 100 95 || 2

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Graph #2.2.12: Histogram for Test Grades

It appears that most of the students had between 60 to 90%. This graph looks somewhat symmetric and also bimodal. The same number of students earned between 60 to 70% and 80 to 90%.

There are other types of graphs for quantitative data. They will be explored in the next section. Section 2.2: Homework 1.) The median incomes of males in each state of the United States, including the

District of Columbia and Puerto Rico, are given in table #2.2.9 (“Median income of,” 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 7 classes. Table #2.2.9: Data of Median Income for Males

\$42,951 \$52,379 \$42,544 \$37,488 \$49,281 \$50,987 \$60,705 \$50,411 \$66,760 \$40,951 \$43,902 \$45,494 \$41,528 \$50,746 \$45,183 \$43,624 \$43,993 \$41,612 \$46,313 \$43,944 \$56,708 \$60,264 \$50,053 \$50,580 \$40,202 \$43,146 \$41,635 \$42,182 \$41,803 \$53,033 \$60,568 \$41,037 \$50,388 \$41,950 \$44,660 \$46,176 \$41,420 \$45,976 \$47,956 \$22,529 \$48,842 \$41,464 \$40,285 \$41,309 \$43,160 \$47,573 \$44,057 \$52,805 \$53,046 \$42,125 \$46,214 \$51,630

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2.) The median incomes of females in each state of the United States, including the District of Columbia and Puerto Rico, are given in table #2.2.10 (“Median income of,” 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 7 classes. Table #2.2.10: Data of Median Income for Females

\$31,862 \$40,550 \$36,048 \$30,752 \$41,817 \$40,236 \$47,476 \$40,500 \$60,332 \$33,823 \$35,438 \$37,242 \$31,238 \$39,150 \$34,023 \$33,745 \$33,269 \$32,684 \$31,844 \$34,599 \$48,748 \$46,185 \$36,931 \$40,416 \$29,548 \$33,865 \$31,067 \$33,424 \$35,484 \$41,021 \$47,155 \$32,316 \$42,113 \$33,459 \$32,462 \$35,746 \$31,274 \$36,027 \$37,089 \$22,117 \$41,412 \$31,330 \$31,329 \$33,184 \$35,301 \$32,843 \$38,177 \$40,969 \$40,993 \$29,688 \$35,890 \$34,381

3.) The density of people per square kilometer for African countries is in table

#2.2.11 (“Density of people,” 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 8 classes.

Table #2.2.11: Data of Density of People per Square Kilometer 15 16 81 3 62 367 42 123 8 9 337 12 29 70 39 83

26 51 79 6 157 105 42 45 72 72 37 4 36 134 12 3

630 563 72 29 3 13 176 341 415 187 65 194 75 16 41 18 69 49 103 65 143 2 18 31

4.) The Affordable Care Act created a market place for individuals to purchase health

care plans. In 2014, the premiums for a 27 year old for the bronze level health insurance are given in table #2.2.12 (“Health insurance marketplace,” 2013). Create a frequency distribution, relative frequency distribution, and cumulative frequency distribution using 5 classes.

Table #2.2.12: Data of Health Insurance Premiums \$114 \$119 \$121 \$125 \$132 \$139 \$139 \$141 \$143 \$145 \$151 \$153 \$156 \$159 \$162 \$163 \$165 \$166 \$170 \$170 \$176 \$177 \$181 \$185 \$185 \$186 \$186 \$189 \$190 \$192 \$196 \$203 \$204 \$219 \$254 \$286

5.) Create a histogram and relative frequency histogram for the data in table #2.2.9.

Describe the shape and any findings you can from the graph.

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6.) Create a histogram and relative frequency histogram for the data in table #2.2.10. Describe the shape and any findings you can from the graph.

7.) Create a histogram and relative frequency histogram for the data in table #2.2.11. Describe the shape and any findings you can from the graph.

8.) Create a histogram and relative frequency histogram for the data in table #2.2.12. Describe the shape and any findings you can from the graph.

9.) Create an ogive for the data in table #2.2.9. Describe any findings you can from the graph.

10.) Create an ogive for the data in table #2.2.10. Describe any findings you can from the graph.

11.) Create an ogive for the data in table #2.2.11. Describe any findings you can from the graph.

12.) Create an ogive for the data in table #2.2.12. Describe any findings you can from the graph.

13.) Students in a statistics class took their first test. The following are the scores they earned. Create a frequency distribution and histogram for the data using class limits that make sense for grade data. Describe the shape of the distribution.

Table #2.2.13: Data of Test 1 Grades 80 79 89 74 73 67 79 93 70 70 76 88 83 73 81 79 80 85 79 80 79 58 93 94 74

14.) Students in a statistics class took their first test. The following are the scores they

earned. Create a frequency distribution and histogram for the data using class limits that make sense for grade data. Describe the shape of the distribution. Compare to the graph in question 13.

Table #2.2.14: Data of Test 1 Grades 67 67 76 47 85 70 87 76 80 72 84 98 84 64 65 82 81 81 88 74 87 83

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Section 2.3: Other Graphical Representations of Data There are many other types of graphs. Some of the more common ones are the frequency polygon, the dot plot, the stem-and-leaf plot, scatter plot, and a time-series plot. There are also many different graphs that have emerged lately for qualitative data. Many are found in publications and websites. The following is a description of the stem-and-leaf plot, the scatter plot, and the time-series plot. Stem-and-Leaf Plots Stem-and-leaf plots are a quick and easy way to look at small samples of numerical data. You can look for any patterns or any strange data values. It is easy to compare two samples using stem-and-leaf plots. The first step is to divide each number into 2 parts, the stem (such as the leftmost digit) and the leaf (such as the rightmost digit). There are no set rules, you just have to look at the data and see what makes sense. Example #2.3.1: Stem-and-Leaf Plot for Grade Distribution

The following are the percentage grades of 25 students from a statistics course. Draw a stem-and-leaf plot of the data.

Table #2.3.1: Data of Test Grades

62 87 81 69 87 62 45 95 76 76 62 71 65 67 72 80 40 77 87 58 84 73 93 64 89

Solution: Divide each number so that the tens digit is the stem and the ones digit is the leaf. 62 becomes 6|2. Make a vertical chart with the stems on the left of a vertical bar. Be sure to fill in any missing stems. In other words, the stems should have equal spacing (for example, count by ones or count by tens). The graph #2.3.1 shows the stems for this example. Graph #2.3.1: Stem-and-Leaf Plot for Test Grades Step 1

4 5 6 7 8 9

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Now go through the list of data and add the leaves. Put each leaf next to its corresponding stem. Don’t worry about order yet just get all the leaves down. When the data value 62 is placed on the plot it looks like the plot in graph #2.3.2. Graph #2.3.2: Stem-and-Leaf Plot for Test Grades Step 2

4 5 6 2 7 8 9

When the data value 87 is placed on the plot it looks like the plot in graph #2.3.3. Graph #2.3.3: Stem-and-Leaf Plot for Test Grades Step 3

4 5 6 2 7 8 7 9

Filling in the rest of the leaves to obtain the plot in graph #2.3.4. Graph #2.3.4: Stem-and-Leaf Plot for Test Grades Step 4

4 5 0 5 8 6 2 9 2 2 5 7 4 7 6 6 1 2 7 3 8 7 1 7 0 7 4 9 9 5 3

Now you have to add labels and make the graph look pretty. You need to add a label and sort the leaves into increasing order. You also need to tell people what the stems and leaves mean by inserting a legend. Be careful to line the leaves up in columns. You need to be able to compare the lengths of the rows when you interpret the graph. The final stem-and-leaf plot for the test grade data is in graph #2.3.5.

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Graph #2.3.5: Stem-and-Leaf Plot for Test Grades Test Scores

4 0 = 40% 4 0 5 5 8 6 2 2 2 4 5 7 9 7 1 2 3 6 6 7 8 0 1 4 7 7 7 9 9 3 5

Now you can interpret the stem-and-leaf display. The data is bimodal and somewhat symmetric. There are no gaps in the data. The center of the distribution is around 70.

Scatter Plot Sometimes you have two different variables and you want to see if they are related in any way. A scatter plot helps you to see what the relationship would look like. A scatter plot is just a plotting of the ordered pairs. Example #2.3.2: Scatter Plot

Is there any relationship between elevation and high temperature on a given day? The following data are the high temperatures at various cities on a single day and the elevation of the city. Table #2.3.2: Data of Temperature versus Elevation

Elevation (in feet) 7000 4000 6000 3000 7000 4500 5000 Temperature (°F) 50 60 48 70 55 55 60

Solution: Preliminary: State the random variables Let x = altitude y = high temperature Now plot the x values on the horizontal axis, and the y values on the vertical axis. Then set up a scale that fits the data on each axes. Once that is done, then just plot the x and y values as an ordered pair.

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Graph #2.3.6: Scatter Plot of Temperature versus Elevation

Looking at the graph, it appears that there is a linear relationship between temperature and elevation. It also appears to be a negative relationship, thus as elevation increases, the temperature decreases.

Time-Series A time-series plot is a graph showing the data measurements in chronological order, the data being quantitative data. For example, a time-series plot is used to show profits over the last 5 years. To create a time-series plot, the time always goes on the horizontal axis, and the other variable goes on the vertical axis. Then plot the ordered pairs and connect the dots. The purpose of a time-series graph is to look for trends over time. Caution, you must realize that the trend may not continue. Just because you see an increase, doesn’t mean the increase will continue forever. As an example, prior to 2007, many people noticed that housing prices were increasing. The belief at the time was that housing prices would continue to increase. However, the housing bubble burst in 2007, and many houses lost value, and haven’t recovered. Example #2.3.3: Time-Series Plot

The following table tracks the weight of a dieter, where the time in months is measuring how long since the person started the diet.

Table #2.3.3: Data of Weights versus Time Time (months) 0 1 2 3 4 5 Weight (pounds) 200 195 192 193 190 187 Make a time-series plot of this data

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Solution: Graph #2.3.7: Time-Series Graph of Weight versus Time

Notice, that over the 5 months, the weight appears to be decreasing. Though it doesn’t look like there is a large decrease.

Be careful when making a graph. If you don’t start the vertical axis at 0, then the change can look much more dramatic than it really is. As an example, graph #2.3.8 shows the graph #2.3.7 with a different scaling on the vertical axis. Notice the decrease in weight looks much larger than it really is.

Graph #2.3.8: Example of a Poor Graph

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Section 2.3: Homework 1.) Students in a statistics class took their first test. The data in table #2.3.4 are the

scores they earned. Create a stem-and-leaf plot.

Table #2.3.4: Data of Test 1 Grades 80 79 89 74 73 67 79 93 70 70 76 88 83 73 81 79 80 85 79 80 79 58 93 94 74

2.) Students in a statistics class took their first test. The data in table #2.3.5 are the

scores they earned. Create a stem-and-leaf plot. Compare to the graph in question 1.

Table #2.3.5: Data of Test 1 Grades 67 67 76 47 85 70 87 76 80 72 84 98 84 64 65 82 81 81 88 74 87 83

3.) When an anthropologist finds skeletal remains, they need to figure out the height

of the person. The height of a person (in cm) and the length of one of their metacarpal bone (in cm) were collected and are in table #2.4.6 (“Prediction of height,” 2013). Create a scatter plot and state if there is a relationship between the height of a person and the length of their metacarpal.

Table #2.3.6: Data of Metacarpal versus Height Length of

Metacarpal Height of

Person 45 171 51 178 39 157 41 163 48 172 49 183 46 173 43 175 47 173

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4.) Table #2.3.7 contains the value of the house and the amount of rental income in a year that the house brings in (“Capital and rental,” 2013). Create a scatter plot and state if there is a relationship between the value of the house and the annual rental income.

Table #2.3.7: Data of House Value versus Rental Value Rental Value Rental Value Rental Value Rental 81000 6656 77000 4576 75000 7280 67500 6864 95000 7904 94000 8736 90000 6240 85000 7072

121000 12064 115000 7904 110000 7072 104000 7904 135000 8320 130000 9776 126000 6240 125000 7904 145000 8320 140000 9568 140000 9152 135000 7488 165000 13312 165000 8528 155000 7488 148000 8320 178000 11856 174000 10400 170000 9568 170000 12688 200000 12272 200000 10608 194000 11232 190000 8320 214000 8528 208000 10400 200000 10400 200000 8320 240000 10192 240000 12064 240000 11648 225000 12480 289000 11648 270000 12896 262000 10192 244500 11232 325000 12480 310000 12480 303000 12272 300000 12480

5.) The World Bank collects information on the life expectancy of a person in each

country (“Life expectancy at,” 2013) and the fertility rate per woman in the country (“Fertility rate,” 2013). The data for 24 randomly selected countries for the year 2011 are in table #2.3.8. Create a scatter plot of the data and state if there appears to be a relationship between life expectancy and the number of births per woman.

Table #2.3.8: Data of Life Expectancy versus Fertility Rate Life

Expectancy Fertility

Rate Life

Expectancy Fertility

Rate 77.2 1.7 72.3 3.9 55.4 5.8 76.0 1.5 69.9 2.2 66.0 4.2 76.4 2.1 55.9 5.2 75.0 1.8 54.4 6.8 78.2 2.0 62.9 4.7 73.0 2.6 78.3 2.1 70.8 2.8 72.1 2.9 82.6 1.4 80.7 1.4 68.9 2.6 74.2 2.5 81.0 1.5 73.3 1.5 54.2 6.9 67.1 2.4

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6.) The World Bank collected data on the percentage of gross domestic product (GDP) that a country spends on health expenditures (“Health expenditure,” 2013) and the percentage of woman receiving prenatal care (“Pregnant woman receiving,” 2013). The data for the countries where this information is available for the year 2011 is in table #2.3.9. Create a scatter plot of the data and state if there appears to be a relationship between percentage spent on health expenditure and the percentage of woman receiving prenatal care.

Table #2.3.9: Data of Prenatal Care versus Health Expenditure Prenatal Care (%)

Health Expenditure (% of GDP)

47.9 9.6 54.6 3.7 93.7 5.2 84.7 5.2

100.0 10.0 42.5 4.7 96.4 4.8 77.1 6.0 58.3 5.4 95.4 4.8 78.0 4.1 93.3 6.0 93.3 9.5 93.7 6.8 89.8 6.1

7.) The Australian Institute of Criminology gathered data on the number of deaths

(per 100,000 people) due to firearms during the period 1983 to 1997 (“Deaths from firearms,” 2013). The data is in table #2.3.10. Create a time-series plot of the data and state any findings you can from the graph.

Table #2.3.10: Data of Year versus Number of Deaths due to Firearms Year 1983 1984 1985 1986 1987 1988 1989 1990 Rate 4.31 4.42 4.52 4.35 4.39 4.21 3.40 3.61 Year 1991 1992 1993 1994 1995 1996 1997 Rate 3.67 3.61 2.98 2.95 2.72 2.95 2.3

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8.) The economic crisis of 2008 affected many countries, though some more than others. Some people in Australia have claimed that Australia wasn’t hurt that badly from the crisis. The bank assets (in billions of Australia dollars (AUD)) of the Reserve Bank of Australia (RBA) for the time period of March 2007 through March 2013 are contained in table #2.3.11 (“B1 assets of,” 2013). Create a time- series plot and interpret any findings.

Table #2.3.11: Data of Date versus RBA Assets

Date

Assets in billions of AUD

Mar-2006 96.9 Jun-2006 107.4 Sep-2006 107.2 Dec-2006 116.2 Mar-2007 123.7 Jun-2007 134.0 Sep-2007 123.0 Dec-2007 93.2 Mar-2008 93.7 Jun-2008 105.6 Sep-2008 101.5 Dec-2008 158.8 Mar-2009 118.7 Jun-2009 111.9 Sep-2009 87.0 Dec-2009 86.1 Mar-2010 83.4 Jun-2010 85.7 Sep-2010 74.8 Dec-2010 76.0 Mar-2011 75.7 Jun-2011 75.9 Sep-2011 75.2 Dec-2011 87.9 Mar-2012 91.0 Jun-2012 90.1 Sep-2012 83.9 Dec-2012 95.8 Mar-2013 90.5

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9.) The consumer price index (CPI) is a measure used by the U.S. government to describe the cost of living. Table #2.3.12 gives the cost of living for the U.S. from the years 1947 through 2011, with the year 1977 being used as the year that all others are compared (DeNavas-Walt, Proctor & Smith, 2012). Create a time- series plot and interpret.

Table #2.3.12: Data of Time versus CPI

Year

CPI-U-RS1 index

(December 1977=100)

Year

CPI-U-RS1 index

(December 1977=100)

1947 37.5 1980 127.1 1948 40.5 1981 139.2 1949 40.0 1982 147.6 1950 40.5 1983 153.9 1951 43.7 1984 160.2 1952 44.5 1985 165.7 1953 44.8 1986 168.7 1954 45.2 1987 174.4 1955 45.0 1988 180.8 1956 45.7 1989 188.6 1957 47.2 1990 198.0 1958 48.5 1991 205.1 1959 48.9 1992 210.3 1960 49.7 1993 215.5 1961 50.2 1994 220.1 1962 50.7 1995 225.4 1963 51.4 1996 231.4 1964 52.1 1997 236.4 1965 52.9 1998 239.7 1966 54.4 1999 244.7 1967 56.1 2000 252.9 1968 58.3 2001 260.0 1969 60.9 2002 264.2 1970 63.9 2003 270.1 1971 66.7 2004 277.4 1972 68.7 2005 286.7 1973 73.0 2006 296.1 1974 80.3 2007 304.5 1975 86.9 2008 316.2 1976 91.9 2009 315.0 1977 97.7 2010 320.2 1978 104.4 2011 330.3 1979 114.4

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10.) The median incomes for all households in the U.S. for the years 1967 to 2011 are given in table #2.3.13 (DeNavas-Walt, Proctor & Smith, 2012). Create a time- series plot and interpret.

Table #2.3.13: Data of Time versus Median Income Year Median Income Year Median Income 1967 42,056 1990 49,950 1968 43,868 1991 48,516 1969 45,499 1992 48,117 1970 45,146 1993 47,884 1971 44,707 1994 48,418 1972 46,622 1995 49,935 1973 47,563 1996 50,661 1974 46,057 1997 51,704 1975 44,851 1998 53,582 1976 45,595 1999 54,932 1977 45,884 2000 54,841 1978 47,659 2001 53,646 1979 47,527 2002 53,019 1980 46,024 2003 52,973 1981 45,260 2004 52,788 1982 45,139 2005 53,371 1983 44,823 2006 53,768 1984 46,215 2007 54,489 1985 47,079 2008 52,546 1986 48,746 2009 52,195 1987 49,358 2010 50,831 1988 49,737 2011 50,054 1989 50,624

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11.) State everything that makes graph #2.3.9 a misleading or poor graph.

Graph #2.3.9: Example of a Poor Graph

12.) State everything that makes graph #2.3.10 a misleading or poor graph (Benen, 2011).

Graph #2.3.10: Example of a Poor Graph

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13.) State everything that makes graph #2.3.11 a misleading or poor graph (“United States unemployment,” 2013).

Graph #2.3.11: Example of a Poor Graph

14.) State everything that makes graph #2.3.12 a misleading or poor graph.

Graph #2.3.12: Example of a Poor Graph

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