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Learning Math: Data Analysis, Statistics, and Probability

Data Organization and Representation Part E: Bar Graphs and Relative Frequencies (30 Minutes)

In This Part: Frequency Bar Graphs

The line plot is a useful graph for examining small sets of data. It’s especially helpful as a device for learning basic statistical ideas. But for larger data sets, it can be awkward to create, since for each data value there is a corresponding dot. That’s a lot of dots for data sets with hundreds or thousands of values! You can, however, replace a line plot with a frequency bar graph.

Let’s look at the transition from line plot to frequency bar graph.

We start with the line plot we’ve been using. Remember that the number of dots over each value on the horizontal axis corresponds to the frequency of that data value:







Now draw a rectangle over each value, with a height corresponding to the frequency of that value:







Now remove the dots, and add a vertical scale that indicates the frequency of each value on the horizontal scale:







The frequency bar graph contains the same information as the line plot for the counts of raisin boxes, but it doesn’t indicate the raisin count for each individual box. The height of each bar or rectangle tells us the frequency for the corresponding raisin count.

In This Part: Relative Frequency

Although the frequency bar graph is useful in many ways, it, like the line plot, can be an awkward graph for large data sets, since the vertical axis corresponds to the frequency of each data value. For large data sets, some data values occur many times and have a high frequency. Consequently, the vertical axis would have to be scaled according to the largest frequency. Imagine the sheet of paper you’d need for the economy-size box of raisins!

An alternative is to use relative frequency, or frequency as a proportion of the whole set. A relative or proportional comparison is usually more useful than a comparison of absolute frequencies. For example, the statement “Five of the 17 boxes have 28 raisins” is more useful than the statement “Five boxes have 28 raisins.”

In this case, the relative frequency of the count 5 is 5/17, which can also be written in decimal form as .294 (rounded to three digits). To find the percentage, multiply the decimal by 100 to obtain 29.4%. This means that 29.4% of the raisin boxes contain 28 raisins.

Here is a frequency table for the raisin count, with the corresponding relative frequencies written as fractions, decimals, and percentages:






Problem E1

Complete the table above. Give decimals to three decimal places and percentages to the nearest tenth of a percent.

Notice that the relative frequencies expressed as fractions add up to 17/17, which equals 1. The relative frequencies expressed as decimals also sum to 1, and the relative frequencies expressed as percentages add up to 100%. The total of the relative frequencies expressed as decimals, however, may not always be exactly 1 due to round-off error; they will occasionally add to 1.002 or 0.997, for example, or something very close to 1. Accordingly, the total percentage may not sum to exactly 100%. To decrease round-off error, we would have to increase the number of decimal places used when rounding.

A relative frequency bar graph looks just like a frequency bar graph except that the units on the vertical axis are expressed as percentages. In the raisin example, the height of each bar is the relative frequency of the corresponding raisin count, expressed as a percentage: See Note 9, below.







One advantage to using relative frequencies is that the total of all relative frequencies in a data set should be 1 (or very close to 1, depending on round-off error), or 100%. In this way, a relative frequency bar graph allows you to think of the data in terms of the whole set in contrast to a frequency bar graph, which only provides you with individual counts.

In This Part: Comparing Representations

In statistics, it can be difficult to provide a specific answer to a question because of the variation present in the data. Statistical analysis allows us to organize data in different ways so that we can draw out potential patterns in the variation and give better answers to the questions posed.

The following Interactive Activity (Flash interactive has been disabled) allows you to review and compare the various representations of data we have explored, both graphical and tabular.

It’s important to note that several kinds of answers can be given when there is variation in your data. Some answers may be stated as intervals, and some answers, like the mode and the median, use a specific value to represent all the different data values.

Video Segment
In this video segment, meteorologist Kim Martucci demonstrates how she solves the statistical problem of predicting the weather. Watch this segment after you have completed Session 2.

What are Kim Martucci’s strategies for predicting the weather? How are they similar to your strategies for counting raisins? How are they different?

You can find the first part of this segment on the session video approximately 23 minutes and 3 seconds after the Annenberg Media logo. The second part of this segment begins approximately 24 minutes and 48 seconds after the Annenberg Media logo.


Note 9

The next transition in the representation is to replace frequencies with relative frequencies. The relative frequency bar graph looks exactly the same as the frequency bar graph. It is important to note that the sizes of the bars remain the same whether you use frequencies or relative frequencies.

If you are working with actual raisins, draw a frequency bar graph and a relative frequency bar graph with your data. Do your graphs look the same?

Groups: Small groups can present their bar graphs to the whole group at this time.


Problem E1

Series Directory

Learning Math: Data Analysis, Statistics, and Probability


Produced by WGBH Educational Foundation. 2001.
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  • ISBN: 1-57680-481-X