A raw score is a data point that has not yet been transformed or analyzed.

A frequency distribution describes the pattern of a set of numbers by displaying a count or proportion for each possible value of a variable.

A frequency table is a visual depiction of data that shows how often each value occurred, that is, how many scores were at each value. Values are listed in one column, and the numbers of individuals with scores at that value are listed in the second column.

MASTERING THE CONCEPT

2.1: A frequency table shows the pattern of the data by indicating how many participants had each possible score. The data in a frequency table can be graphed in a frequency histogram or a frequency polygon.

EXAMPLE 2.1

The most popular sport in the world is what Canadians and Americans call soccer (called football by most people in the world). A recent book analyzes soccer from the perspectives of several social sciences—statistics, economics, psychology, geography, and sociology. In Soccernomics (Kuper & Szymanski, 2009), the authors explore fascinating social science questions and present data about where soccer is most popular. Using data on percentages of soccer spectators out of the entire population, they conclude that soccer is most popular in England, followed by Spain, Germany, Italy, and France (in that order). But we wondered: Does popularity coincide with success?

World Cup Powerhouses Enthusiasm for women’s soccer in the United States continued to grow after the U.S. team’s gold-medal victory at the 2012 Olympic Games, especially after the team lost to an inspired Japanese team in the 2011 World Cup. Like Olympic gold, winning the World Cup is a career-capping ambition. Frequency data from the FIFA website (through 2011) show that some countries have far more top finishes than others. For example, West Germany/Germany had 10 first- or second-place finishes and Brazil had 8. These two teams are shown here playing each other in the 2007 Women’s World Cup.

AP Photo/Anja Niedringhaus

Table 2-3 depicts data from the World Cup Web site (http://www.fifa.com), listing the years in which countries came in first or second in the tournament. The table is in alphabetical order by country. Of the 80 countries that have participated in at least one men’s or women’s World Cup tournament, only 16 countries have placed first or second, an indication that some countries dominate. The remaining 64 countries have never finished in first or second place. We can use these data to create a frequency table.

At first glance, it is not easy to find a pattern in most lists of numbers. But when we reorder those numbers, a pattern begins to emerge. A frequency table is the best way to create an easy-to-understand distribution of data. In this example, we simply organize the data into a table with two columns, one for the range of responses (the values) and one for the frequencies of each response (the scores).

There are specific steps to create a frequency table. First, we determine the range of raw scores. For each country, we can count how many first- or second-place finishes these countries have had: 4, 8, 1, 2, 1, 2, 2, 6, 1, 2, 1, 2, 3, 3, 2, and 10. In addition, 64 countries had 0 first- or second-place finishes. We know at a glance that the lowest score is 0. A quick glance also reveals that the highest score is 10; one country finished in first or second place in 10 World Cup tournaments, a most impressive number. Simply noting that the scores range from 0 to 10 brings some clarity to the data set. But we can do even better.

After we identify the lowest and highest scores, we create the two columns that we see in Table 2-4. We examine the raw scores and determine how many countries fall at each value in the range. The appropriate number for each value is recorded in the table. For example, there is one country with 10 first- or second-place finishes, so a 1 is marked there. It is important to note that we include all numbers in the range; there are no countries with 9, 7, or 5 top finishes, so we put a 0 next to each one.

Here is a recap of the steps to create a frequency table:

1. Determine the highest score and the lowest score.
2. Create two columns; label the first with the variable name, and label the second “Frequency.”
3. List the full range of values that encompasses all the scores in the data set, from highest to lowest. Include all values in the range, even those for which the frequency is 0.
4. Count the number of scores at each value, and write those numbers in the frequency column.

As shown in Table 2-5, we can add a column for percentages. To calculate a percentage, we divide the number of countries at a certain value by the total number of countries, and then multiply by 100. As we observed earlier, 1 out of 80 countries had 10 top finishes.

So, for the score of 10 top finishes, the percentage for 1 of 80 countries is 1.25%.

Note that when we calculate statistics, we can come up with different answers depending on the number of steps and how we decide to round numbers. In this book, we round off to three decimal places throughout the calculations, but we report the final answers to two decimal places, rounding up or down as appropriate. Sometimes the numbers don’t add up to 100% exactly, due to rounding. But if you follow this guideline, then you should get the same answers that we get.

Creating a frequency table for the data gives us more insight into the set of numbers. We can see that two countries, Brazil and West Germany/Germany, are well above the others. Indeed, the subtitle for Soccernomics includes the phrase Why Germany and Brazil Win. What about England, the country in which soccer is most popular? It’s been one of the top two finishers only once, when it won in 1966. It seems clear that the popularity of the sport doesn’t necessarily relate to World Cup success.

A grouped frequency table is a visual depiction of data that reports frequencies within a given interval rather than the frequencies for a specific value.

EXAMPLE 2.2

The following data exemplify the first of these two situations in which the data aren’t easily conveyed in a standard frequency table. These are the pacing indices, to two decimal places, for the 87 television shows, some of which are listed in Table 2-1.

A quick glance at these data does not really tell us the pacing index of the typical television show. A frequency table wouldn’t be helpful either. The lowest score is 14.95 and the highest is 56.90. So the table would include 14.95, 14.96, 14.97, and so on, all the way to 56.90! Such a table would be absurdly long and would not convey much more useful information than does the list of the original raw data.

Instead of reporting every single value in the range, we can report intervals, or ranges of values. Here are the five steps to generate a standard grouped frequency table:

STEP 1: Find the highest and lowest scores in the frequency distribution.

In the pacing index example, these scores are 56.90 and 14.95.

STEP 2: Get the full range of data.

If there are decimal places, round both the highest and the lowest scores down to the nearest whole numbers. If they already are whole numbers, use these. Subtract the lowest whole number from the highest whole number and add 1 to get the full range of the data. (Why do we add 1? Try it yourself. If we subtract 14 from 56, we get 42—but count the values from 14 through 56, including the numbers at either end. There are 43 numbers, and we want to know the full range of the data.)

In the pacing index example, 14.95 and 56.90 round down to 14 and 56, respectively; 56 − 14 = 42, and 42 + 1 = 43. The scores fall within a range of 43.

STEP 3: Determine the number of intervals and the best interval size.

There is no consensus about the ideal number of intervals, but most researchers recommend between 5 and 10 intervals, unless the data set is enormous and has a huge range. To find the best interval size, we divide the range by the number of intervals we want, then round to the nearest whole number (as long as the numbers are not too small—that is, as long as there are not too many decimal places). For ranges that are wide, the size of intervals could be a multiple of 10 or 100 or 1000; for smaller ranges, it could be as small as 2, 3, or 5, or even less than 1, if the numbers go to many decimal places. Try several interval sizes to determine the best one.

In the pacing index example, we might choose to have about 9 intervals. If we choose 9, the interval size will be 5.

STEP 4: Figure out the number that will be the bottom of the lowest interval.

We want the bottom of that interval to be a multiple of the interval size. For example, if we have 9 intervals of size 5, then we want the bottom interval to start at a multiple of 5. It could start at 0, 10, 55, or 105, depending on the data. We select the multiple of 5 that is below the lowest score.

In the pacing index example, there are 9 intervals of size 5, so the bottom of the lowest interval should be a multiple of 5. The lowest score is 14.95, so the bottom of the lowest interval would be 10. (If the lowest score were 7.22, we would choose 5. Note that this process might lead to one more interval than we planned for; this is perfectly fine. In our case, we have 10, rather than the 9 intervals we had estimated.)

STEP 5: Finish the table by listing the intervals from highest to lowest and then counting the numbers of scores in each.

This step is much like creating a frequency table (without intervals), which we discussed earlier. If we decide on intervals of size 5 and the first one begins at 10, then we count the five numbers that fall in this interval: 10, 11, 12, 13, and 14. The interval in this example runs from 10 to 14. (In reality, it runs from 10 to 14.9999, and the next one begins at 15, five digits higher than the bottom of the preceding interval.) A good rule of thumb is that the bottom of the intervals should jump by the chosen interval size, in this case 5.

In the pacing index example, the lowest interval would be 10 to 14, or 10.00 to 14.99. The next one would be 15.00 to 19.99, and so on.

The grouped frequency table in Table 2-6 gives us a much better sense of the pacing indices of the TV shows in this sample than either the list of raw data or a frequency table without intervals.

MASTERING THE CONCEPT

2.2: The data in a frequency table can be viewed in graph form. In a histogram, bars are used to depict frequencies at each score or interval. In a frequency polygon, a dot is placed above each score or interval to indicate the frequency, and the dots are connected.

A histogram is a graph that looks like a bar graph but depicts just one variable, usually based on scale data, with the values of the variable on the x-axis and the frequencies on the y-axis.

EXAMPLE 2.3

Let’s start by constructing a histogram from a frequency table. Table 2-4 depicts data on countries’ numbers of World Cup top finishes. We construct a histogram by drawing the x-axis (horizontal) and y-axis (vertical) of a graph. We label the x-axis with the variable of interest—in our case, “First- or second-place finishes”—and we label the y-axis “Frequency.” As with most graphs, the lowest numbers start where the axes intersect and the numbers go up as we go to the right on the x-axis and as we go up on the y-axis. Ideally, the lowest number on each axis is 0, so that the graphs are not misleading. However, if the range of numbers on either axis is far from 0, histograms sometimes use a number other than 0 as the lowest number. Further, if there are negative numbers among the scores (as can be the case in scores for air temperature, for example), the x-axis could have negative numbers.

Once we’ve created the graph, we draw a bar for each value. Each bar is centered on the value for which it provides the frequency. The height of the bars represents the numbers of scores that fall at each value—the frequencies. If no country had a score at a particular value, then we would not draw a bar for that value. So, for the value of 2 on the x-axis, a bar centers on 2 with a height of 6 on the y-axis, indicating that six countries had a first- or second-place finish twice. Figure 2-1 shows the histogram for the World Cup data.

Figure 2-1

Histogram for the Frequency Table of World Cup Successes Histograms are graphic depictions of the information in frequency tables or grouped frequency tables. This histogram shows how many countries had a certain number of first- or second-place finishes in the men’s and women’s World Cup soccer tournaments through 2011.

Here is a recap of the steps to construct a histogram from a frequency table:

1. Draw the x-axis and label it with the variable of interest and the full range of values for this variable. (Include 0 unless all of the scores are so far from 0 that this would be impractical.)
2. Draw the y-axis, label it “Frequency,” and include the full range of frequencies for this variable. (Include 0 unless it’s impractical.)
3. Draw a bar for each value, centering the bar around that value on the x-axis and drawing the bar as high as the frequency for that value, as represented on the y-axis.

Grouped frequency tables can also be depicted as histograms. Instead of listing values on the x-axis, we list the midpoints of intervals. Students commonly make mistakes in determining midpoints. If an interval ranges from 0 to 9, what is the mid- point? If you said 4.5, you’re making a very common mistake. Remember, this interval really goes from 0.000000 to 9.999999, or as close as you can get to 10, the bottom of the next interval, without actually being 10. Given that there are 10 numbers in this range (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), the midpoint would be 5 from the bottom. So the midpoint for 0 to 9 is 5. A good rule: When determining a midpoint, look at the bottom of the interval that you’re interested in and then the bottom of the next interval; then, determine the midpoint of these two numbers.

EXAMPLE 2.4

Let’s look at the TV pacing index data for which we constructed a grouped frequency histogram. What are the midpoints of the 10 intervals? Let’s calculate the midpoint for the lowest interval, 10 to 14.99. We should look at the bottom of this interval, 10.00, and the bottom of the next interval, 15.00. The midpoint of these numbers is 12.50, so that is the midpoint of this interval. The remaining midpoints can be calculated the same way. For the highest interval, 55.00 to 59.99, it helps to imagine that we had one more interval. If we did, it would start at 60.00. The midpoint of 55.00 and 60.00 is 57.50. Using these guidelines, we calculate the midpoints as 12.50, 17.50, 22.50, 27.50, 32.50, 37.50, 42.50, 47.50, 52.50, and 57.50. (A good check is to see if the midpoints should jump by the interval size—in this case, 5.) We now can construct the histogram by placing these midpoints on the x-axis and drawing bars that center on them and are as high as the frequency for each interval. The histogram for these data is shown in Figure 2-2.

Here is a recap of the steps to construct a histogram from a grouped frequency table:

1. Determine the midpoint for every interval.
2. Draw the x-axis, label it with the variable of interest and with the midpoints for each interval on this variable. (Include 0 unless it’s impractical.)
3. Draw the y-axis, label it “Frequency,” and include the full range of frequencies for this variable. (Include 0 unless it’s impractical.)
4. Draw a bar for each midpoint, centering the bar on that midpoint on the x-axis and drawing the bar as high as the frequency for that interval, as represented on the y-axis.

A frequency polygon is a line graph, with the x-axis representing values (or midpoints of intervals) and the y-axis representing frequencies; a dot is placed at the frequency for each value (or midpoint), and the dots are connected.

EXAMPLE 2.5

For the most part, we make frequency polygons exactly as we make histograms. Instead of constructing bars above each value or midpoint, however, we draw dots and connect them. The other difference is that we need to add an appropriate value (or midpoint) on either end of the graph so that we can draw lines down to 0, grounding the shape. In the case of the TV pacing data, we calculate one more midpoint on each end by subtracting the interval size, 5, from the bottom midpoint (12.50 − 5 = 7.5) and adding the interval size, 5, to the top midpoint (57.5 + 5 = 62.5). We now can construct the frequency polygon by placing these midpoints on the x-axis, drawing dots at each midpoint that are as high as the frequency for each interval, and connecting the dots. Figure 2-3 shows the frequency polygon for the grouped frequency distribution of TV pacing indices that we constructed previously in Figure 2-2.

Here is a recap of the steps to construct a frequency polygon. When basing a frequency polygon on a frequency table, we place the specific values on the x-axis. When basing it on a grouped frequency table, we place the midpoints of intervals on the x-axis.

1. If based on a grouped frequency table, determine the midpoint for every interval. If based on a frequency table, skip this step.
2. Draw the x-axis and label it with the variable of interest and either the values or the midpoints. (Include 0 unless it’s impractical.)
3. Draw the y-axis, label it “Frequency,” and include the full range of frequencies for this variable. (Include 0 unless it’s impractical.)
4. Mark a dot above each value or midpoint depicting the frequency, as represented on the y-axis, for that value or midpoint, and connect the dots.
5. Add an appropriate hypothetical value or midpoint on both ends of the x-axis, and mark a dot for a frequency of 0 for each of these values or midpoints. Connect the existing line to these dots to create a shape rather than a “floating” line.

CHECK YOUR LEARNING

Clarifying the Concepts

• 2-1 Name four different ways to organize raw scores visually.
• 2-2 What is the difference between frequencies and grouped frequencies?

Calculating the Statistics

• 2-3 In 2013, U.S. News & World Report published a list of the citations per faculty member score for the 400 best universities in the world. As examples, the Massachusetts Institute of Technology is number 1 on the list of best universities and McGill University in Montreal is number 18. The citation score tells us how many times a faculty member’s research has been cited by other researchers over the previous 5 years, and is an indicator of research productivity. Here are the data for the top 50 institutions:
  
  
  1. Construct a grouped frequency table of these data.
  2. Construct a histogram for this grouped frequency table.
  3. Construct a frequency polygon for this grouped frequency table.

Applying the Concepts

• 2-4 Consider the data from Check Your Learning 2-3, as well as the table and graphs that you constructed.
  1. What can we tell from the graphs and table that we cannot tell from a quick glance at the list of scores?
  2. What issues might arise in considering data across countries with a wide range of academic systems?