Relative Frequency Bar Charts

From Counts to Relative Frequencies

In Reading a Bar Chart of Counts, you used bar heights to find how many individuals or items belonged to each category. A count is useful, but it depends on the size of the group being described. If one group has more individuals than another, it may have higher counts in several categories simply because it is larger. A relative frequency bar chart displays proportions instead, making it easier to compare distributions from groups of different sizes.

To make the change, divide each category’s count by the total number of individuals represented. The denominator must match the group whose distribution you want to describe. The result is the fraction, decimal, or percentage of that group in the category. As covered in Frequency Tables and Relative Frequency, relative frequencies summarize how the total is divided among categories.

For example, if 12 of 30 people in a group choose a particular category, its relative frequency is \(12/30=0.40\), or \(40\%\). The proportion and percentage communicate the same information in different forms. Keep the form consistent when labeling a graph: a decimal scale might run from 0 to 1, while a percentage scale might run from 0% to 100%.

Making and Checking the Graph

A relative frequency bar chart has the same basic structure as a count bar chart: categories appear on the horizontal axis, and separated bars represent them. The key change is the vertical axis. It shows relative frequency or percent rather than count. Each bar’s height is the category’s relative frequency in the group.

To construct the chart, first find the total number of individuals or items. Divide each category count by that same total. Then label the axes with the variable and the relative frequency scale, and draw one bar for each category to the calculated height. As with a count chart, bars have equal widths and gaps between them. Start the vertical scale at zero so the bar heights represent the proportions accurately.

The total check follows from dividing all the counts by the same total: the counts add to the group total, so their relative frequencies add to 1. If rounded percentages add to 99.9% or 100.1%, that small difference may be due to rounding. A much larger mismatch suggests a calculation or denominator error.

Worked Example: A Relative Frequency Bar Chart

Why Relative Frequency Helps Compare Groups

Counts and relative frequencies answer different questions. A count tells how many individuals in a group belong to a category. A relative frequency tells what fraction of that group belongs to the category. When two groups have different totals, comparing their counts alone can be misleading: the larger group may have more individuals in a category even when the category represents a smaller share of that group.

To compare the distributions, calculate each group’s relative frequencies using its own total. Do not divide the counts in both groups by the grand total unless you specifically want to describe each group-category combination as a share of everyone combined. For comparing what is typical within each group, use that group’s total as the denominator.

Worked Example: Comparing Two Groups of Different Sizes

Reading the Bars and Choosing the Right Comparison

On a relative frequency chart, the tallest bar identifies the category with the largest share of the group, not necessarily the largest count across several groups. Always identify the group represented by a bar and note the scale. If a bar reaches 0.30 on a decimal scale, it represents 30% of the relevant group. To find an approximate count from a relative frequency, multiply that proportion by the group total; the chart alone does not give the count unless the total is also known.

A comparison between two percentages is often stated as a percentage-point difference. For instance, if one group has 65% in a category and another has 45%, subtract to get \(20\) percentage points. As discussed in Percent Versus Percentage Point Differences, that subtraction is not the same as a relative percent change. For a basic comparison of relative frequency bars, naming the two percentages and their percentage-point difference is usually clear.

Relative frequency charts are particularly helpful when groups have different sizes, but they do not make sample sizes irrelevant. A percentage based on a small group can change substantially when just a few individuals are added or removed. Include group sizes when they are available, especially if the purpose is to assess how much information supports the displayed percentages.

Common Mistakes and What a Strong Answer Includes

• Using the wrong denominator. For a group’s relative frequency, divide by that group’s total. In a comparison of two group distributions, each group gets its own denominator.
• Comparing counts as if the groups were the same size. Counts tell how many; relative frequencies tell what share. For different-sized groups, use proportions or percentages to compare the distributions.
• Confusing the vertical scale with a count scale. A bar at 0.45 means 45% of the relevant group, not 0.45 individuals. State whether the axis is in decimals or percentages.
• Forgetting the total check. Relative frequencies for a complete distribution should sum to 1, or 100%. If they do not, recheck the counts, denominators, and arithmetic; allow for small rounding differences.
• Describing a percentage-point difference as a percent change. Subtracting two percentages gives a difference in percentage points. Use that wording when comparing the displayed group percentages.
• Leaving out the groups and categories. A full comparison names which groups are being compared, the category of interest, and the percentages or percentage-point difference in context.

A strong response might say: “Cycling was selected by 65% of the morning group and 45% of the afternoon group, a difference of 20 percentage points. The morning group had the larger cycling share.” This reports the within-group percentages, names the category and groups, and describes the direction of the comparison. If counts are also relevant, report them separately and identify them as counts.