A relative frequency indicates how often a specific kind of event occurs within the total number of observations. It is a type of frequency that uses percentages, proportions, and fractions.
In this post, learn about relative frequencies, the relative frequency distribution, and its cumulative counterpart.
Frequencies vs. Relative Frequencies
A frequency is a count of a particular event. For example, Jim read ten statistics books this year. The football team won 12 games.
In contrast, relative frequencies do not use raw counts. Instead, they relate the count for a particular type of event to the total number of events using percentages, proportions, or fractions. That’s where the term “relative” comes in—a specific tally relative to the total number. For instance, 25% of the books Jim read were about statistics. The football team won 85% of its games.
If you see a count, it’s a frequency. If you see a percentage, proportion, ratio, or fraction, it’s a relative frequency.
Relative frequencies help you place a type of event into a larger context. For example, a survey indicates that 20 students like their statistics course the most. From this raw count, you don’t know if that’s a large or small proportion. However, if you knew that 30 out of 40 (75%) respondents indicated that statistics was their favorite, you’d consider it a high number!
Additionally, they allow you to compare values between studies. Imagine that different sized schools surveyed their students and obtained different numbers of respondents. If 30 students indicate that statistics is their favorite, that could be a high percentage in one school but a low percentage in another, depending on the total number of responses.
Relative frequencies facilitate apples-to-apples comparisons.
How to Find a Relative Frequency
To calculate relative frequencies, you must know both of the following:
- The count of events for a category.
- The total number of events.
Relative frequency calculations convert counts into percentages by taking the count of a specific type of event and dividing it by the total number of observations. Its formula is the following:
For example, imagine a school surveys 50 students and asks them to name their favorite course. Thirty-six students state that statistics is their favorite.
- The frequency of “statistics” responses is 36.
- The total number of responses is 50.
To find the relative frequency for the statistics course, perform the following division: 36 / 50 = 72%.
Relative Frequencies as Empirical Probabilities
Relative frequencies also serve as empirical probabilities. Probabilities define the likelihood of events occurring. Probability calculations often rely heavily on theory. However, when you observe the relative frequency of an event, it’s an empirical probability. In other words, analysts calculate them using real-world observations rather than theory.
An empirical probability is the number of events out of the total number of observations.
Related post: Probability Fundamentals
Relative Frequency Distributions: Tables and Graphs
A relative frequency distribution describes the relative frequencies for all possible outcomes in a study. While a single value is for one type of event, the distribution displays percentages for all possible results. Analysts typically present these distributions using tables and bar charts.
Let’s bring them to life by working through an example!
The relative frequency distribution table below displays the percentage of students in each grade at a small school with 88 students.
|School Grade||Count of Students||Relative Frequency|
If the table had only the first two columns, grade level and count of students, it would be a frequency distribution. A frequency distribution describes the counts for all possible outcomes. It’s the percentage column that makes it a relative frequency distribution. You can see how the two types of distributions are related.
To create a relative frequency distribution table, take the count of students in a row (one grade level) and divide it by the total number of students. For example, in the first row, there are 23 students in the first grade—23 out of 88 = 26.1%. For second graders, it’s 20 out of 88 = 22.7% Repeat this process for all rows in the table.
Because these tables consider all possible outcomes, the total percentage must sum to 100%, excepting rounding error.
They are handy because you instantly know the percentage of the total for each outcome, and you can identify trends and patterns. For example, first graders account for just over a quarter (26.1%) of the entire school by themselves. Conversely, 6th graders make up only 9.1% of the school. There’s a downward trend in values as grade levels increase.
Bar chart example
You can also use bar charts to display relative frequency distributions. The graph below depicts the same information as the table. It shows a clear trend for the upper grades to have smaller percentages of total students.
Related post: Bar Charts: Using, Examples, and Interpreting
Cumulative Relative Frequency Distributions
A cumulative relative frequency distribution sums the progression of relative frequencies through all the possible outcomes. Creating this type of distribution entails adding one more column to the table and summing the values as you move down the rows to create a running, cumulative total.
For this example, we’ll return to school students. The cumulative relative frequency table below adds the final column.
|School Grade||Count of Students||Relative Frequency||Cumulative Relative Frequency|
To find the cumulative value for each row, sum the relative frequencies as you work your way down the rows. The first value in the cumulative row equals that row’s relative frequency. For the 2nd row, add that row’s value to the previous row. In the table, we add 26.1 + 22.7 = 48.8%. In the third row, add 17% to the previous cumulative value, 17 + 48.8 = 65.8%. And so on through all the rows.
The final cumulative value must equal 1 or 100%, excepting rounding error.
You can also display cumulative relative frequency distributions on graphs. In the chart below, I added the orange cumulative line. Use these cumulative distributions to determine where most of the events/observations occur. In the example data, the first and second graders comprise about half the school.
To learn about functions that describe distributions, read my post, Understanding Probability Distributions.