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Section2.1Frequency Tables and Bar Graphs

SubsectionCategorizing Data

Once we have gathered data, we might wish to classify it. Roughly speaking, data can be classified as categorical data or quantitative data.

Quantitative and categorical data

Categorical (qualitative) data are pieces of information that allow us to classify the objects under investigation into various categories.

Quantitative data are responses that are numerical in nature and with which we can perform meaningful arithmetic calculations.

Example2.1

We might conduct a survey to determine the name of the favorite movie that each person in a math class saw in a movie theater.

When we conduct such a survey, the responses would look like: Finding Nemo, The Hulk, or Terminator 3: Rise of the Machines. We might count the number of people who give each answer, but the answers themselves do not have any numerical values: we cannot perform computations with an answer like "Finding Nemo." This would be categorical data.

Example2.2

A survey could ask the number of movies you have seen in a movie theater in the past 12 months (0, 1, 2, 3, 4, ...)

This would be quantitative data.

Other examples of quantitative data would be the running time of the movie you saw most recently (104 minutes, 137 minutes, 104 minutes, ...) or the amount of money you paid for a movie ticket the last time you went to a movie theater ($5.50,$7.75, $9, ...). Sometimes, determining whether or not data is categorical or quantitative can be a bit trickier. Example2.3 Suppose we gather respondents' ZIP codes in a survey to track their geographical location. ZIP codes are numbers, but we can't do any meaningful mathematical calculations with them (it doesn't make sense to say that 98036 is "twice" 49018 that's like saying that Lynnwood, WA is "twice" Battle Creek, MI, which doesn't make sense at all), so ZIP codes are really categorical data. Example2.4 A survey about the movie you most recently attended includes the question "How would you rate the movie you just saw?" with these possible answers:  1 - it was awful 2 - it was just OK 3 - I liked it 4 - it was great 5 - best movie ever! Again, there are numbers associated with the responses, but we can't really do any calculations with them: a movie that rates a 4 is not necessarily twice as good as a movie that rates a 2, whatever that means; if two people see the movie and one of them thinks it stinks and the other thinks it's the best ever it doesn't necessarily make sense to say that "on average they liked it." As we study movie-going habits and preferences, we shouldn't forget to specify the population under consideration. If we survey 3-7 year-olds the runaway favorite might be Finding Nemo. 13-17 year-olds might prefer Terminator 3. And 33-37 year-olds might prefer...well, Finding Nemo. Exploration2.1 Classify each measurement as categorical or quantitative 1. Eye color of a group of people 2. Daily high temperature of a city over several weeks 3. Annual income Solution 1. Categorical 2. Quantitative 3. Quantitative SubsectionPresenting Categorical Data Graphically 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. Frequency Table A frequency table is a table with two 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). Example2.5 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.  Color Frequency Blue 25 Green 52 Red 41 White 36 Black 39 Grey 23 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. Bar graph 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. Example2.6 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: In this case, our chart might benefit from being reordered from largest to smallest frequency values. This arrangement can make it easier to compare similar values in the chart, even without gridlines. When we arrange the categories in decreasing frequency order like this, it is called a Pareto chart. Example2.7 Transforming our bar graph from earlier into a Pareto chart, we get: Example2.8 In a survey1Gallup Poll. March 5-8, 2009. http://www.pollingreport.com/enviro.html, adults were asked whether they personally worried about a variety of environmental concerns. The numbers (out of 1012 surveyed) who indicated that they worried a great deal about some selected concerns are summarized below.  Environmental Issue Frequency Pollution of Drinking Water 597 Contamination of soil and water by toxic waste 526 Air pollution 455 Global warming 354 This data could be shown graphically in a bar graph: To show relative sizes, it is common to use a pie chart. 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. Example2.9 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. Example2.10 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. Exploration2.2 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 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. Here is another way that fanciness can lead to trouble. Instead of plain bars, it is tempting to substitute meaningful images. This type of graph is called a pictogram. Pictogram A pictogram is a statistical graphic in which the size of the picture is intended to represent the frequencies or size of the values being represented. Example2.11 A labor union might produce the graph to the right to show the difference between the average manager salary and the average worker salary. Looking at the picture, it would be reasonable to guess that the manager salaries is 4 times as large as the worker salaries the area of the bag looks about 4 times as large. However, the manager salaries are in fact only twice as large as worker salaries, which were reflected in the picture by making the manager bag twice as tall. 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. Example2.12 Compare the two graphs below showing support for same-sex marriage rights from a poll taken in December 2008 . 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. Exploration2.3 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. Solution 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. SubsectionPresenting Quantitative Data Graphically Quantitative, or numerical, data can also be summarized into frequency tables. Example2.13 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:  Score Frequency 0 2 5 1 12 1 15 2 16 2 17 4 18 8 19 4 20 6 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. Histogram A histogram is like a bar graph, but where the horizontal axis is a number line. Example2.14 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 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 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. Example2.15 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.  Interval Frequency 120-134 4 135-149 14 150-164 16 165-179 28 180-194 12 195-209 8 210-224 7 225-239 6 240-254 2 255-269 3 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. Exploration2.4 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 Solution Using a class intervals of size 55, we can group our data into six intervals:  Cost interval Frequency$140-194 5 $195-249 3$250-304 9 $305-359 12$360-414 4 \$415-469 3

We can use the frequency distribution to generate the histogram.