14.1: Organizing and Visualizing Data
Categorical, or qualitative, data are pieces of information that allow us to classify the objects under investigation into various categories. We usually begin working with categorical data by summarizing the data into a frequency table.
A frequency table is a table with two or three columns. One column lists the categories, and another for the frequencies with which the items in the categories occur (how many items fit into each category). The last column is the relative frequencies that give the percent of the total.
Frequency: Number of times a data value occurs in a data set.
Frequency Distribution: A listing of each data value or grouping of data values (called classes) with their frequencies.
Relative Frequency: The frequency divided by n, the size of the sample. This gives the percent of the total for each data value or class of data values.
Relative Frequency Distribution: A listing of each data value or class of data values with their relative frequencies
An insurance company determines vehicle insurance premiums based on known risk factors. If a person is considered a higher risk, their premiums will be higher. One potential factor is the color of your car. The insurance company believes that people with some color cars are more likely to get in accidents. To research this, they examine police reports for recent total-loss collisions. The data is summarized in the frequency table below.
\(\begin{array}{|l|l|}
\hline \textbf { Color } & \textbf { Frequency } \\
\hline \text { Blue } & 25 \\
\hline \text { Green } & 52 \\
\hline \text { Red } & 41 \\
\hline \text { White } & 36 \\
\hline \text { Black } & 39 \\
\hline \text { Grey } & 23 \\
\hline
\end{array}\)
Sometimes we need an even more intuitive way of displaying data. This is where charts and graphs come in. There are many, many ways of displaying data graphically, but we will concentrate on one very useful type of graph called a bar graph. In this section we will work with bar graphs that display categorical data; the next section will be devoted to bar graphs that display quantitative data.
A bar graph is a graph that displays a bar for each category with the length of each bar indicating the frequency of that category.
To construct a bar graph, we need to draw a vertical axis and a horizontal axis. The vertical direction will have a scale and measure the frequency of each category; the horizontal axis has no scale in this instance. The construction of a bar chart is most easily described by use of an example.
Using our car data from above, note the highest frequency is 52, so our vertical axis needs to go from 0 to 52, but we might as well use 0 to 55, so that we can put a hash mark every 5 units:
Notice that the height of each bar is determined by the frequency of the corresponding color. The horizontal gridlines are a nice touch, but not necessary. In practice, you will find it useful to draw bar graphs using graph paper, so the gridlines will already be in place, or using technology. Instead of gridlines, we might also list the frequencies at the top of each bar, like this:
To show relative sizes, it is common to use a pie chart.
A pie chart is a circle with wedges cut of varying sizes marked out like slices of pie or pizza. The relative sizes of the wedges correspond to the relative frequencies of the categories.
For our vehicle color data, a pie chart might look like this:
Pie charts can often benefit from including frequencies or relative frequencies (percents) in the chart next to the pie slices. Often having the category names next to the pie slices also makes the chart clearer.
The pie chart to the right shows the percentage of voters supporting each candidate running for a local senate seat.
If there are 20,000 voters in the district, the pie chart shows that about 11% of those, about 2,200 voters, support Reeves.
Pie charts look nice, but are harder to draw by hand than bar charts since to draw them accurately we would need to compute the angle each wedge cuts out of the circle, then measure the angle with a protractor. Computers are much better suited to drawing pie charts. Common software programs like Microsoft Word or Excel, OpenOffice.org Write or Calc, or Google Docs are able to create bar graphs, pie charts, and other graph types. There are also numerous online tools that can create graphs[2].
Create a bar graph and a pie chart to illustrate the grades on a history exam below.
A: 12 students, B: 19 students, C: 14 students, D: 4 students, F: 5 students
- Answer
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Don’t get fancy with graphs! People sometimes add features to graphs that don’t help to convey their information. For example, 3-dimensional bar charts like the one shown below are usually not as effective as their two-dimensional counterparts.
Another distortion in bar charts results from setting the baseline to a value other than zero. The baseline is the bottom of the vertical axis, representing the least number of cases that could have occurred in a category. Normally, this number should be zero.
Compare the two graphs below showing support for same-sex marriage rights from a poll taken in December 2008[3]. The difference in the vertical scale on the first graph suggests a different story than the true differences in percentages; the second graph makes it look like twice as many people oppose marriage rights as support it.
A poll was taken asking people if they agreed with the positions of the 4 candidates for a county office. Does the pie chart present a good representation of this data? Explain.
- Answer
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While the pie chart accurately depicts the relative size of the people agreeing with each candidate, the chart is confusing, since usually percents on a pie chart represent the percentage of the pie the slice represents.
Quantitative, or numerical, data can also be summarized into frequency tables.
A teacher records scores on a 20-point quiz for the 30 students in his class. The scores are:
19 20 18 18 17 18 19 17 20 18 20 16 20 15 17 12 18 19 18 19 17 20 18 16 15 18 20 5 0 0
These scores could be summarized into a frequency table by grouping like values:
\(\begin{array}{|c|c|}
\hline \textbf { Score } & \textbf { Frequency } \\
\hline 0 & 2 \\
\hline 5 & 1 \\
\hline 12 & 1 \\
\hline 15 & 2 \\
\hline 16 & 2 \\
\hline 17 & 4 \\
\hline 18 & 8 \\
\hline 19 & 4 \\
\hline 20 & 6 \\
\hline
\end{array}\)
Using this table, it would be possible to create a standard bar chart from this summary, like we did for categorical data:
However, since the scores are numerical values, this chart doesn’t really make sense; the first and second bars are five values apart, while the later bars are only one value apart. It would be more correct to treat the horizontal axis as a number line. This type of graph is called a histogram .
A histogram is like a bar graph, but where the horizontal axis is a number line
For the values above, a histogram would look like:
Notice that in the histogram, a bar represents values on the horizontal axis from that on the left hand-side of the bar up to, but not including, the value on the right hand side of the bar. Some people choose to have bars start at ½ values to avoid this ambiguity.
Unfortunately, not a lot of common software packages can correctly graph a histogram. About the best you can do in Excel or Word is a bar graph with no gap between the bars and spacing added to simulate a numerical horizontal axis.
If we have a large number of widely varying data values, creating a frequency table that lists every possible value as a category would lead to an exceptionally long frequency table, and probably would not reveal any patterns. For this reason, it is common with quantitative data to group data into class intervals .
Class intervals are groupings of the data. In general, we define class intervals so that:
- Each interval is equal in size. For example, if the first class contains values from 120-129, the second class should include values from 130-139.
- We have somewhere between 5 and 20 classes, typically, depending upon the number of data we’re working with.
Suppose that we have collected weights from 100 male subjects as part of a nutrition study. For our weight data, we have values ranging from a low of 121 pounds to a high of 263 pounds, giving a total span of 263-121 = 142. We could create 7 intervals with a width of around 20, 14 intervals with a width of around 10, or somewhere in between. Often time we have to experiment with a few possibilities to find something that represents the data well. Let us try using an interval width of 15. We could start at 121, or at 120 since it is a nice round number.
\(\begin{array}{|c|c|}
\hline \textbf { Interval } & \textbf { Frequency } \\
\hline 120-134 & 4 \\
\hline 135-149 & 14 \\
\hline 150-164 & 16 \\
\hline 165-179 & 28 \\
\hline 180-194 & 12 \\
\hline 195-209 & 8 \\
\hline 210-224 & 7 \\
\hline 225-239 & 6 \\
\hline 240-254 & 2 \\
\hline 255-269 & 3 \\
\hline
\end{array}\)
A histogram of this data would look like:
In many software packages, you can create a graph similar to a histogram by putting the class intervals as the labels on a bar chart.
Other graph types such as pie charts are possible for quantitative data. The usefulness of different graph types will vary depending upon the number of intervals and the type of data being represented. For example, a pie chart of our weight data is difficult to read because of the quantity of intervals we used.
The total cost of textbooks for the term was collected from 36 students. Create a histogram for this data.
$140 $160 $160 $165 $180 $220 $235 $240 $250 $260 $280 $285
$285 $285 $290 $300 $300 $305 $310 $310 $315 $315 $320 $320
$330 $340 $345 $350 $355 $360 $360 $380 $395 $420 $460 $460
- Answer
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Using a class intervals of size 55, we can group our data into six intervals:
\(\begin{array}{|l|r|}
\hline \textbf { cost interval } & \textbf { Frequency } \\
\hline \$ 140-194 & 5 \\
\hline \$ 195-249 & 3 \\
\hline \$ 250-304 & 9 \\
\hline \$ 305-359 & 12 \\
\hline \$ 360-414 & 4 \\
\hline \$ 415-469 & 3 \\
\hline
\end{array}\)We can use the frequency distribution to generate the histogram.
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.
- Answer
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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
- Answer
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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 4 7 8 8 9 | |
| 8 | 0 1 3 5 8 | |
| 9 | 0 0 3 3 |
[1] Gallup Poll. March 5-8, 2009. http://www.pollingreport.com/enviro.htm
[2] For example: http://nces.ed.gov/nceskids/createAgraph/ or http://docs.google.com
[3]CNN/Opinion Research Corporation Poll. Dec 19-21, 2008, from http://www.pollingreport.com/civil.htm