## 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:

### Notes

**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.

### Solutions

**Problem E1**