Session 2, Part E:
Bar Graphs and Relative Frequencies

In This Part: Frequency Bar Graphs | Relative Frequency | Comparing Representations

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:

 Raisin Count Frequency Relative Frequency Fraction Decimal %
 25 1 1/17 .059 5.9 26 2 2/17 .118 27 3 17.6 28 5 29 4 .235 30 1 1/17 31 1

Problem E1

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

 Raisin Count Frequency Relative Frequency Fraction Decimal %
 25 1 1/17 .059 5.9 26 2 2/17 .118 11.8 27 3 3/17 .176 17.6 28 5 5/17 .294 29.4 29 4 4/17 .235 23.5 30 1 1/17 .059 5.9 31 1 1/17 .059 5.9

 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:  Note 9 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.

 Session 2: Index | Notes | Solutions | Video

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