2.4: Other Types of Graphs
- Page ID
- 10920
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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 or Google Sheets can make both of them.
Frequency or Relative Frequency Tables
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. The classes for categorical frequency tables are typically found in the data you have collected. For example, if you collected data where the question was 'what is your favorite color?' then your table might have one class for each color if the answers aren't too varied. However, you may choose to use a more compact set of categories such as 'shades of blue,' 'shades of red,' etc. if there colors were open ended. Or, you might choose to make categories only representing the most selected values with an 'other' category for anything else. How you choose to structure your table will just depend on what you are studying and how best the summary will reflect your findings.
Suppose you have the following data for which type of car students at a college drive. How can we analyze the data?
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
Solution:
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:
| Category | Frequency |
|---|---|
| Ford | 5 |
| Chevy | 12 |
| Honda | 6 |
| Toyota | 12 |
| Nissan | 10 |
| Other | 5 |
| Total | 50 |
Table 2.1.1: Frequency Table for Type of Car Data
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.
Remember: Relative frequency =\(\frac{f}{n}\) where \(f\) is the frequency of the class and \(n\) is the total number of data values. If needed, rounding to at least three decimal places is suggested. Relative frequency is just a way of looking at the proportion of the data that is in each category, so another way of labeling the relative frequency category could be Proportion of Cars.
As an example for Ford category:
relative frequency \(= \frac{5}{50} = 0.10\)
This can be written as a decimal, fraction, or percent. You now have a relative frequency distribution:
| 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 |
Table 2.1.2: Relative Frequency Table for Type of Car Data
The relative frequency column should add up to 1.00. It might be off a little due to rounding errors.
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
Bar graphs, also called Bar 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.
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.
Draw a bar graph of the data in Example 2.1.1.
Solution:
| 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 |
Table 2.1.2: Relative Frequency Table for Type of Car Data
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.
You can also use MS Excel or Google Sheets to create a bar graph from the frequency table.
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Graph 2.1.1: Bar Graph 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.
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.
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Graph 2.1.2: Relative Frequency Bar Graph for Type of Car Data
The population in Park City is made up of children, working-age adults, and retirees. Table shows the three age groups, the number of people in the town from each age group, and the proportion (%) of people in each age group. Remember, relative frequency is just the proportion of the total number of data values in each category. So the column labeled Proportion of population is just relative frequency as a percentage.
Construct a bar graph showing the proportions.
| Age groups | Number of people | Proportion of population |
|---|---|---|
| Children | 67,059 | 19% |
| Working-age adults | 152,198 | 43% |
| Retirees | 131,662 | 38% |
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Figure \(\PageIndex{4}\): This is a bar graph that matches the supplied data. The x-axis shows age groups, and the y-axis shows the percentages of Park City's population.
Park city is broken down into six voting districts. The table shows the percent of the total registered voter population that lives in each district as well as the percent total of the entire population that lives in each district. Construct a bar graph that shows the registered voter population by district.
| District | Registered voter population | Overall city population |
|---|---|---|
| 1 | 15.5% | 19.4% |
| 2 | 12.2% | 15.6% |
| 3 | 9.8% | 9.0% |
| 4 | 17.4% | 18.5% |
| 5 | 22.8% | 20.7% |
| 6 | 22.3% | 16.8% |
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Figure \(\PageIndex{6}\): This is a bar graph that matches the supplied data. The x-axis shows Park City voting districts, and the y-axis shows the percentages of the registered voter population.
Pie Charts
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.
Draw a pie chart of the data in Example 2.1.1.
First you need the relative frequencies.
| 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 |
Table 2.1.2: Relative Frequency Table for Type of Car Data
Solution:
Then you multiply each relative frequency by 360° to obtain the angle measure for each category. As with most intermediate calculation steps, using an unrounded relative frequency is the best practice when calculating degrees.
As an example for Ford category:
\(0.10 \cdot 360^{\circ}=36.0^{\circ}\)
| 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 |
Table 2.1.3: Pie Chart Angles for Type of Car Data
Now draw the pie chart using a compass, protractor, and straight edge, if asked to do so by hand. To simplify the process, it is suggested to start with a circle labeled with 0°, 90°, 180°, and 270° and then approximate the pie slices by cumulatively adding angles as you travel around the circle back to 360°.
However, technology is preferred. If you use technology, there is no need for the relative frequencies or the angles. Technology like MS Excel or Google Sheets will create pie charts very quickly.
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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 are 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.
Pareto Charts
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.
Other Graphs
Multiple Bar Graph
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.
In the Wii Fit game, you can do four different types of 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 yoga 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.
Dot Plot
Dot plots can be used to display various types of information. Figure \(\PageIndex{1}\) uses a dot plot to display the number of M & M's of each color found in a bag of M & M's. Each dot represents a single M & M. From the figure, you can see that there were \(3\) blue M & M's, \(19\) brown M & M's, etc.
Dot plots can also be used for Quantitative data where their are few categories, such as the number of siblings or the age of high school graduates. These dot plots, since their horizontal axis positioning is based on a real number line can also help determine the distribution or shape of the data.
When the number of data values is large than is reasonable to plot a dot to represent each one, then a variation on the dot plot is used. The dot plot in Figure \(\PageIndex{2}\) shows the number of people playing various card games on the Yahoo website on a Wednesday. Unlike Figure \(\PageIndex{1}\), the location rather than the number of dots represents the frequency.
The dot plot in Figure \(\PageIndex{3}\) shows the number of people playing on a Sunday and on a Wednesday. This graph makes it easy to compare the popularity of the games separately for the two days, but does not make it easy to compare the popularity of a given game on the two days.
The dot plot in Figure \(\PageIndex{4}\) makes it easy to compare the days of the week for specific games while still portraying differences among games.
More Qualitative Data Graphs
Stem-and-Leaf Graph (Stemplots)
One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.
For Susan Dean's spring pre-calculus class, scores for the first exam were as follows (smallest to largest):
33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100
| Stem | Leaf |
|---|---|
| 3 | 3 |
| 4 | 2 9 9 |
| 5 | 3 5 5 |
| 6 | 1 3 7 8 8 9 9 |
| 7 | 2 3 4 8 |
| 8 | 0 3 8 8 8 |
| 9 | 0 2 4 4 4 4 6 |
| 10 | 0 |
The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% \(\left(\frac{8}{31}\right)\) were in the 90s or 100, a fairly high number of As.
For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest):
32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; 61
Construct a stem plot for the data.
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Stem Leaf 3 2 2 3 4 8 4 0 2 2 3 4 6 7 7 8 8 8 9 5 0 0 1 2 2 2 3 4 6 7 7 6 0 1
The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later.
The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data:
1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3
Do the data seem to have any concentration of values?
HINT: The leaves are to the right of the decimal.
Answer
The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers.
| Stem | Leaf |
|---|---|
| 1 | 1 5 |
| 2 | 3 5 7 |
| 3 | 2 3 3 5 8 |
| 4 | 0 2 5 5 7 8 |
| 5 | 5 6 |
| 6 | 5 7 |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 11 | |
| 12 | 3 |
The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers:
0.5; 0.7; 1.1; 1.2; 1.2; 1.3; 1.3; 1.5; 1.5; 1.7; 1.7; 1.8; 1.9; 2.0; 2.2; 2.5; 2.6; 2.8; 2.8; 2.8; 3.5; 3.8; 4.4; 4.8; 4.9; 5.2; 5.5; 5.7; 5.8; 8.0
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Stem Leaf 0 5 7 1 1 2 2 3 3 5 5 7 7 8 9 2 0 2 5 6 8 8 8 3 5 8 4 4 8 9 5 2 5 7 8 6 7 8 0 The value 8.0 may be an outlier. Values appear to concentrate at one and two miles.
A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. Tables \(\PageIndex{1}\) and \(\PageIndex{2}\) show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.
| President | Ageat Inauguration | President | Age | President | Age |
|---|---|---|---|---|---|
| Pierce | 48 | Harding | 55 | Obama | 47 |
| Polk | 49 | T. Roosevelt | 42 | G.H.W. Bush | 64 |
| Fillmore | 50 | Wilson | 56 | G. W. Bush | 54 |
| Tyler | 51 | McKinley | 54 | Reagan | 69 |
| Van Buren | 54 | B. Harrison | 55 | Ford | 61 |
| Washington | 57 | Lincoln | 52 | Hoover | 54 |
| Jefferson | 57 | Grant | 46 | Truman | 60 |
| Madison | 57 | Hayes | 54 | Eisenhower | 62 |
| J. Q. Adams | 57 | Arthur | 51 | L. Johnson | 55 |
| Monroe | 58 | Garfield | 49 | Kennedy | 43 |
| J. Adams | 61 | A. Johnson | 56 | F. Roosevelt | 51 |
| Jackson | 61 | Cleveland | 47 | Nixon | 56 |
| Taylor | 64 | Taft | 51 | Clinton | 47 |
| Buchanan | 65 | Coolidge | 51 | Trump | 70 |
| W. H. Harrison | 68 | Cleveland | 55 | Carter | 52 |
| President | Age | President | Age | President | Age |
|---|---|---|---|---|---|
| Washington | 67 | Lincoln | 56 | Hoover | 90 |
| J. Adams | 90 | A. Johnson | 66 | F. Roosevelt | 63 |
| Jefferson | 83 | Grant | 63 | Truman | 88 |
| Madison | 85 | Hayes | 70 | Eisenhower | 78 |
| Monroe | 73 | Garfield | 49 | Kennedy | 46 |
| J. Q. Adams | 80 | Arthur | 56 | L. Johnson | 64 |
| Jackson | 78 | Cleveland | 71 | Nixon | 81 |
| Van Buren | 79 | B. Harrison | 67 | Ford | 93 |
| W. H. Harrison | 68 | Cleveland | 71 | Reagan | 93 |
| Tyler | 71 | McKinley | 58 | ||
| Polk | 53 | T. Roosevelt | 60 | ||
| Taylor | 65 | Taft | 72 | ||
| Fillmore | 74 | Wilson | 67 | ||
| Pierce | 64 | Harding | 57 | ||
| Buchanan | 77 | Coolidge | 60 |
Answer
| Ages at Inauguration | Ages at Death | |
|---|---|---|
| 9 9 8 7 7 7 6 3 2 | 4 | 6 9 |
| 8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 1 1 1 1 1 0 | 5 | 3 6 6 7 7 8 |
| 9 5 4 4 2 1 1 1 0 | 6 | 0 0 3 3 4 4 5 6 7 7 7 8 |
| 7 | 0 0 1 1 1 3 4 7 8 8 9 | |
| 8 | 0 1 3 5 8 | |
| 9 | 0 0 3 3 |
The table shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stem-and-leaf plot of these wins and losses.
| Losses | Wins | Year | Losses | Wins | Year |
|---|---|---|---|---|---|
| 34 | 48 | 1968–1969 | 41 | 41 | 1989–1990 |
| 34 | 48 | 1969–1970 | 39 | 43 | 1990–1991 |
| 46 | 36 | 1970–1971 | 44 | 38 | 1991–1992 |
| 46 | 36 | 1971–1972 | 39 | 43 | 1992–1993 |
| 36 | 46 | 1972–1973 | 25 | 57 | 1993–1994 |
| 47 | 35 | 1973–1974 | 40 | 42 | 1994–1995 |
| 51 | 31 | 1974–1975 | 36 | 46 | 1995–1996 |
| 53 | 29 | 1975–1976 | 26 | 56 | 1996–1997 |
| 51 | 31 | 1976–1977 | 32 | 50 | 1997–1998 |
| 41 | 41 | 1977–1978 | 19 | 31 | 1998–1999 |
| 36 | 46 | 1978–1979 | 54 | 28 | 1999–2000 |
| 32 | 50 | 1979–1980 | 57 | 25 | 2000–2001 |
| 51 | 31 | 1980–1981 | 49 | 33 | 2001–2002 |
| 40 | 42 | 1981–1982 | 47 | 35 | 2002–2003 |
| 39 | 43 | 1982–1983 | 54 | 28 | 2003–2004 |
| 42 | 40 | 1983–1984 | 69 | 13 | 2004–2005 |
| 48 | 34 | 1984–1985 | 56 | 26 | 2005–2006 |
| 32 | 50 | 1985–1986 | 52 | 30 | 2006–2007 |
| 25 | 57 | 1986–1987 | 45 | 37 | 2007–2008 |
| 32 | 50 | 1987–1988 | 35 | 47 | 2008–2009 |
| 30 | 52 | 1988–1989 | 29 | 53 | 2009–2010 |
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Table \(\PageIndex{3}\): Atlanta Hawks Wins and Losses Number of Wins Number of Losses 3 1 9 9 8 8 6 5 2 5 5 9 8 7 6 6 5 5 4 3 1 1 1 1 0 3 0 2 2 2 2 4 4 5 6 6 6 9 9 9 8 8 7 6 6 6 3 3 3 2 2 1 1 0 4 0 0 1 1 2 4 5 6 6 7 7 8 9 7 7 6 3 2 0 0 0 0 5 1 1 1 2 3 4 4 6 7 6 9
Line Graph
Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in Example, the x-axis (horizontal axis) consists of data values and the y-axis (vertical axis) consists of frequency points. The frequency points are connected using line segments.
In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores. The results are shown in Table and in Figure.
| Number of times teenager is reminded | Frequency |
|---|---|
| 0 | 2 |
| 1 | 5 |
| 2 | 8 |
| 3 | 14 |
| 4 | 7 |
| 5 | 4 |
Answer
In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table. Construct a line graph.
| Number of times in shop | Frequency |
|---|---|
| 0 | 7 |
| 1 | 10 |
| 2 | 14 |
| 3 | 9 |
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Figure \(\PageIndex{2}\): A line graph showing the number of times a car is in the shop on the x-axis and frequency on the y-axis.
Summary
For quantitavie Data we can include a couple more graphs to our tool box. A stem-and-leaf plot is a way to plot data and look at the distribution. In a stem-and-leaf plot, all data values within a class are visible. The advantage in a stem-and-leaf plot is that all values are listed, unlike a histogram, which gives classes of data values. A line graph is often used to represent a set of data values in which a quantity varies with time. These graphs are useful for finding trends. That is, finding a general pattern in data sets including temperature, sales, employment, company profit or cost over a period of time. For qualitative data, we have bar graphs, pie charts and a few more. A bar graph is a chart that uses either horizontal or vertical bars to show comparisons among categories. One axis of the chart shows the specific categories being compared, and the other axis represents a discrete value. Some bar graphs present bars clustered in groups of more than one (grouped bar graphs), and others show the bars divided into subparts to show cumulative effect (stacked bar graphs). Bar graphs are especially useful when categorical data is being used.
References
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- “9th Annual AP Report to the Nation.” CollegeBoard, 2013. Available online at http://apreport.collegeboard.org/goa...omoting-equity (accessed September 13, 2013).
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