Frequency vs relative frequency tables

Frequency vs relative frequency: the short answer

A frequency is a count: how many observations fall in a category. A relative frequency is that same count divided by the total number of observations, which turns it into a proportion between 0 and 1, often written as a percent.

They carry the same information, so you can always convert one into the other. The reason both exist is comparison: raw counts tell you how many, while relative frequencies tell you what share, and only the share lets you compare groups that are not the same size.

What a frequency is

A frequency table shows the number of observational units in each category of a categorical variable. If 40 students are surveyed and 24 walk to school, the frequency for walking is 24. Frequencies are whole counts, they are never negative, and adding them across every category gives the total number of observations.

Frequencies are the natural output of tallying data, and they answer how-many questions directly. Their limit is comparison: a count of 24 means something very different in a class of 40 than in a school of 400, and the raw number alone does not tell you which.

What a relative frequency is

A relative frequency table shows the proportion of observational units in each category. You compute it as the category frequency divided by the total: $\text{relative frequency} = \frac{\text{count in category}}{\text{total count}}$. For 24 walkers out of 40 students, the relative frequency is $\frac{24}{40} = 0.6$, or 60%.

Because it is a proportion, a relative frequency always lands between 0 and 1, and the relative frequencies across all categories add up to 1, or 100%. The AP course notes that percentages, relative frequencies, and ratios all convey the same information as proportions, so 0.6, 60%, and 3 out of 5 are three ways to say the same thing.

Frequency vs relative frequency side by side

When to use which

Use frequencies when the actual counts matter: how many people were affected, how many defects were found, how many students chose an option. Counts are also what you need before you can compute anything else.

Use relative frequencies when you are comparing categories or groups, especially groups of different sizes. If one school surveys 40 students and another surveys 200, comparing raw counts is misleading, because the larger school will tend to have larger counts everywhere. Converting to proportions puts both on the same 0 to 1 scale so the comparison is fair.

The classic mix-up and how to avoid it

The classic error is comparing raw frequencies across groups of different sizes and drawing a conclusion from the counts alone. A school with 90 walkers out of 200 has more walkers than a school with 24 out of 40, yet the second school has the higher share: $\frac{24}{40} = 0.6$ beats $\frac{90}{200} = 0.45$. The bigger count hides the smaller proportion.

Two checks prevent it. First, before comparing across groups, convert counts to relative frequencies so every group is on the same scale. Second, sanity-check your relative frequencies by adding them: within one group they must total 1, or 100%, and if they do not, a count or the total is wrong.

Where this fits in AP Statistics

Frequency and relative frequency tables appear at the start of Unit 1, in topic 1.3 on tabular representation and summary statistics for one categorical variable. Bar charts and pie charts, covered next, can display either counts or proportions, and you should be ready to read both, as in histogram vs bar graph.

The idea returns in Unit 2 with two-way tables, where a joint relative frequency is a cell count divided by the table total and a marginal relative frequency is a row or column total divided by the table total. Knowing whether a table shows counts or proportions is the first thing to check, and it connects to telling qualitative from quantitative data in the first place.