Relative frequency tells you what portion of a dataset belongs to a particular value or category. To calculate it, divide the frequency of that value by the total number of observations.
Relative frequency formula:
[
\text{Relative Frequency}=\frac{\text{Frequency}}{\text{Total Number of Observations}}
]
For example, if 12 out of 40 students prefer mathematics, the relative frequency is:
[
\frac{12}{40}=0.30=30%
]
The answer means that 30% of the students in the group prefer mathematics.
You can express relative frequency as a fraction, decimal, or percentage. The calculation stays the same; only the way you write the result changes. The important part is using the correct frequency and dividing it by the total number of observations, not by the number of categories.
What Is Relative Frequency?
Relative frequency is the proportion of all observations that fall into a particular category or value. It shows how common something is within a dataset rather than simply telling you how many times it occurred.
Consider a class of 40 students. If 12 students choose mathematics as their favorite subject, the frequency is 12. The relative frequency is 12 divided by 40, which equals 0.30, or 30%.
This distinction matters when comparing groups of different sizes. For example, 15 students choosing a particular subject in a class of 30 represents 50% of the class. In a school of 150 students, 30 choosing the same subject represents only 20%. Although the second group has twice as many students choosing that subject, the first group has the higher relative frequency.
Frequency vs. Relative Frequency
| Measure | What it tells you | Example |
|---|---|---|
| Frequency | The number of times a value or category occurs | 12 students choose mathematics |
| Relative frequency | The proportion of observations in that category | 12 ÷ 40 = 0.30 |
| Percentage frequency | The same proportion expressed as a percentage | 0.30 × 100 = 30% |
Frequency answers the question, “How many?” Relative frequency answers, “What share of the total?”
Relative Frequency Formula
The standard formula is:
[
RF=\frac{f}{n}
]
Where:
- (RF) is the relative frequency.
- (f) is the frequency of the selected value or category.
- (n) is the total number of observations.
If the question asks for a percentage, use:
[
RF(%)=\frac{f}{n}\times100
]
For instance, suppose a survey records 18 people who travel to work by bus out of 60 respondents.
[
RF=\frac{18}{60}=0.30
]
To convert the decimal to a percentage:
[
0.30\times100=30%
]
The relative frequency is 0.30, or 30%.
One detail is easy to overlook: the denominator represents the entire group being studied. It is not the number of different answers, the largest frequency, or the number of rows in a table.
How to Calculate Relative Frequency Step by Step
Follow these steps when you have a count, a list of observations, or a frequency table.
Step 1: Identify the Category or Value
Read the question carefully and identify exactly what you need to calculate.
If a survey asks how many students prefer football, for example, you need the count for football. You should not use the count for another sport or combine categories unless the question specifically asks for a combined result.
Step 2: Find the Frequency
The frequency is the number of observations that belong to the category you selected.
Suppose 9 students choose football. Your frequency is:
[
f=9
]
If the question gives you a frequency table, find the number in the relevant row. If it provides raw data, count how often the value appears.
Step 3: Find the Total Number of Observations
Add all the category frequencies or count every observation in the dataset.
Suppose 9 students choose football, 11 choose basketball, and 10 choose volleyball. The total is:
[
n=9+11+10=30
]
Use 30 as the denominator for each category because all three categories belong to the same group of students.
Step 4: Divide the Frequency by the Total
For football, divide its frequency of 9 by the total of 30.
[
RF=\frac{9}{30}=0.30
]
The relative frequency of students who choose football is 0.30.
Step 5: Convert the Result if Necessary
If the question requests a percentage, multiply the decimal by 100.
[
0.30\times100=30%
]
If it asks for a fraction, you can write (9/30), simplified to (3/10). If it asks for a decimal, write 0.30.
Always follow the requested answer format. A correct decimal can still be marked wrong if the question specifically asks for a percentage and you do not convert it.
Step 6: Check Your Answer
For an individual category, relative frequency must be between 0 and 1, inclusive. As a percentage, it must be between 0% and 100%.
If you calculate relative frequencies for every category in a complete table, their total should equal 1, or 100%. This check helps reveal an incorrect denominator, a counting mistake, or a missing category.
How to Calculate Relative Frequency From a Frequency Table
A frequency table already provides the count for each category, so you do not need to count the original observations again. Your task is to divide each row’s frequency by the total frequency.
Suppose 40 students were asked to name their favorite school subject.
| Favorite subject | Frequency | Relative frequency | Percentage |
|---|---|---|---|
| Mathematics | 12 | 12 ÷ 40 = 0.30 | 30% |
| Science | 8 | 8 ÷ 40 = 0.20 | 20% |
| English | 10 | 10 ÷ 40 = 0.25 | 25% |
| History | 10 | 10 ÷ 40 = 0.25 | 25% |
| Total | 40 | 1.00 | 100% |
Here is how to read the table:
Mathematics: Divide 12 by 40 to get 0.30, or 30%.
Science: Divide 8 by 40 to get 0.20, or 20%.
English: Divide 10 by 40 to get 0.25, or 25%.
History: Divide 10 by 40 to get 0.25, or 25%.
Notice that the denominator stays at 40 for every row. You are measuring each category as a share of the same group, so you should not divide by the frequency in the row above or by the number of categories.
What If the Frequencies Do Not Add Up to the Stated Total?
Check the table before calculating. Suppose the question says there are 40 students, but the frequencies add up to 38. Do not automatically use 38 or 40 without investigating.
There may be a missing category, an omitted response, or a transcription error. If the table is incomplete, the calculated proportions may not describe the full group. Use the denominator specified by the problem only when the listed frequencies are consistent with it, or follow any instructions about missing responses.
How to Calculate Relative Frequency From Raw Data
Sometimes a question gives you a list of values rather than a completed frequency table. In that case, count how often each value appears before applying the formula.
Suppose 10 students report how many books they read last month:
2, 1, 3, 2, 0, 2, 1, 3, 2, 4
There are 10 observations. Count the number of times each distinct value appears.
| Books read | Frequency | Relative frequency | Percentage |
|---|---|---|---|
| 0 | 1 | 1 ÷ 10 = 0.10 | 10% |
| 1 | 2 | 2 ÷ 10 = 0.20 | 20% |
| 2 | 4 | 4 ÷ 10 = 0.40 | 40% |
| 3 | 2 | 2 ÷ 10 = 0.20 | 20% |
| 4 | 1 | 1 ÷ 10 = 0.10 | 10% |
| Total | 10 | 1.00 | 100% |
The value 2 appears four times, so its frequency is 4. Dividing by the 10 observations gives a relative frequency of 0.40, or 40%.
The most common counting error is forgetting repeated values or counting a value twice. If the list is long, arrange the values in order or use tally marks to keep track of each occurrence. Then add the frequency column to confirm that it matches the number of observations in the original list.
How to Calculate Relative Frequency as a Percentage
A relative frequency and a percentage describe the same proportion in different formats. To convert a decimal relative frequency to a percentage, multiply by 100 and add the percent sign.
| Frequency | Total | Decimal relative frequency | Percentage |
|---|---|---|---|
| 6 | 24 | 0.25 | 25% |
| 7 | 20 | 0.35 | 35% |
| 3 | 16 | 0.1875 | 18.75% |
| 9 | 40 | 0.225 | 22.5% |
Take the third row as an example:
[
\frac{3}{16}=0.1875
]
Multiply by 100:
[
0.1875\times100=18.75%
]
If your teacher requests two decimal places for the decimal answer, 0.1875 rounds to 0.19. If the requested answer is a percentage to two decimal places, the result is 18.75%.
Do not round too early. Keeping the original fraction or full decimal until the final step reduces rounding errors, particularly when completing an entire table.
How to Calculate Relative Frequency for Grouped Data
Grouped data places observations into intervals instead of listing every individual value. For example, a test score table might use the intervals 0 to 9, 10 to 19, 20 to 29, and 30 to 39.
The same formula applies:
[
RF=\frac{\text{Frequency of the interval}}{\text{Total frequency}}
]
Suppose the scores of 50 students are grouped as follows.
| Test score | Frequency | Relative frequency | Percentage |
|---|---|---|---|
| 0 to 9 | 4 | 4 ÷ 50 = 0.08 | 8% |
| 10 to 19 | 9 | 9 ÷ 50 = 0.18 | 18% |
| 20 to 29 | 17 | 17 ÷ 50 = 0.34 | 34% |
| 30 to 39 | 20 | 20 ÷ 50 = 0.40 | 40% |
| Total | 50 | 1.00 | 100% |
The interval 20 to 29 contains 17 students. Its relative frequency is (17/50=0.34), or 34%.
If the question asks for the relative frequency of scores from 10 to 29, combine the frequencies for 10 to 19 and 20 to 29 first:
[
RF=\frac{9+17}{50}=\frac{26}{50}=0.52
]
Therefore, 52% of the students scored from 10 to 29.
The important distinction is that the numerator must represent all observations included in the requested interval or range. Do not divide by the number of intervals in the table.
How to Calculate Cumulative Relative Frequency
Cumulative relative frequency shows the proportion of observations that fall in a category or below it in an ordered table. It is different from ordinary relative frequency, which measures one category at a time.
To calculate it, add the frequencies from the first row through the current row, then divide that running total by the total number of observations.
Consider the grouped test scores above.
| Test score | Frequency | Cumulative frequency | Cumulative relative frequency |
|---|---|---|---|
| 0 to 9 | 4 | 4 | 4 ÷ 50 = 0.08 |
| 10 to 19 | 9 | 13 | 13 ÷ 50 = 0.26 |
| 20 to 29 | 17 | 30 | 30 ÷ 50 = 0.60 |
| 30 to 39 | 20 | 50 | 50 ÷ 50 = 1.00 |
By the end of the 20 to 29 interval, 30 students have scores below 30. Dividing 30 by 50 gives a cumulative relative frequency of 0.60, or 60%.
Cumulative relative frequency is useful when a question asks what proportion of observations fall below a value or within a range extending from the beginning of the distribution. The final cumulative relative frequency should be 1, or 100%, when the table includes every observation. It should not decrease as you move down an ordered table.
How to Check Whether Your Relative Frequency Is Correct
Before submitting an answer, use these checks to catch common errors.
1. Check the denominator. It should represent the total number of observations in the group being studied. For a complete frequency table, add the frequencies to find the total.
2. Check the range. A single relative frequency cannot be less than 0 or greater than 1. In percentage form, it cannot be below 0% or above 100%.
3. Add the relative frequencies. If the categories are mutually exclusive and account for every observation, the decimal values should total 1, or the percentages should total 100%.
4. Check the percentage conversion. Multiply the decimal by 100. For example, 0.42 is 42%, not 4.2% or 420%.
5. Consider rounding. Rounded entries may add to 0.99 or 1.01 rather than exactly 1.00. This can happen even when each calculation is correct. Check the unrounded values before changing any entries.
6. Confirm the categories are complete. If observations are missing or a response can appear in more than one category, the total may not behave as expected. Make sure each observation is counted appropriately for the question.
These checks are especially useful when a table has many rows. One incorrect denominator can affect every result, so verify the total before recalculating each category.
Common Relative Frequency Mistakes
| Mistake | Correct approach |
|---|---|
| Dividing by the number of categories | Divide by the total number of observations |
| Using the wrong category’s count | Use the frequency for the exact value or group asked about |
| Treating frequency as relative frequency | Frequency is a count; relative frequency is a proportion |
| Multiplying by 100 before dividing | Divide first, then multiply by 100 if a percentage is required |
| Rounding every step | Keep full precision until the final answer |
| Confusing relative and cumulative relative frequency | Cumulative relative frequency includes the preceding categories in an ordered table |
| Assuming relative frequency always predicts future outcomes | It describes the observed data; future outcomes may differ |
A useful way to avoid most of these mistakes is to write down the numerator and denominator before doing the division. If you can explain what each number represents, you are less likely to use the wrong values.
Worked Relative Frequency Examples
Example 1: Finding Relative Frequency From a Count
Question: In a survey of 25 students, 10 say science is their favorite subject. What is the relative frequency?
The frequency is 10, and the total is 25.
[
RF=\frac{10}{25}=0.40
]
As a percentage:
[
0.40\times100=40%
]
Answer: 0.40, or 40%.
Example 2: Finding Relative Frequency in a Table
Question: A category appears 9 times in a dataset containing 36 observations. Find its relative frequency.
[
RF=\frac{9}{36}=0.25
]
Answer: 0.25, or 25%.
The category represents one quarter of all observations.
Example 3: Finding a Missing Frequency
Question: A dataset contains 50 observations. Three categories have frequencies of 12, 15, and 15. The fourth category is missing. Find its frequency and relative frequency.
First, add the known frequencies:
[
12+15+15=42
]
Subtract the known total from the overall total:
[
50-42=8
]
The missing frequency is 8. Now divide by the total:
[
RF=\frac{8}{50}=0.16
]
Answer: The missing frequency is 8, and its relative frequency is 0.16, or 16%.
This method works when the categories are complete and the stated total includes all observations.
Example 4: Calculating a Percentage With Rounding
Question: A particular response occurs 3 times in a group of 16. Find its relative frequency as a percentage, rounded to two decimal places.
[
RF=\frac{3}{16}=0.1875
]
Convert to a percentage:
[
0.1875\times100=18.75%
]
Answer: 18.75%.
In this case, the percentage is exact to two decimal places, so no additional rounding is needed.
Example 5: Finding Relative Frequency for a Range
Question: In a group of 50 students, 9 scored between 10 and 19, and 17 scored between 20 and 29. Find the relative frequency of students scoring from 10 to 29.
Combine the two relevant frequencies:
[
9+17=26
]
Divide by the total number of students:
[
RF=\frac{26}{50}=0.52
]
Answer: 0.52, or 52%.
Because the question asks about a range covering two intervals, both frequencies belong in the numerator.
Practice Questions
Try these questions before looking at the solutions.
Question 1: In a class of 25 students, 10 prefer science. What is the relative frequency as a decimal and a percentage?
Question 2: A value occurs 9 times in a dataset of 36 observations. Find its relative frequency.
Question 3: A frequency table contains four categories with frequencies of 4, 6, 5, and 5. What is the relative frequency of the category with a frequency of 6?
Answers and Solutions
Answer 1:
[
\frac{10}{25}=0.40=40%
]
Answer 2:
[
\frac{9}{36}=0.25=25%
]
Answer 3:
First find the total frequency:
[
4+6+5+5=20
]
Then divide the selected category’s frequency by the total:
[
\frac{6}{20}=0.30=30%
]
If your answers differ, check that you used the total number of observations as the denominator.
Final Takeaway
To calculate relative frequency, divide the count for the category you are studying by the total number of observations. Convert the decimal to a percentage only when the question asks for it. When working with a frequency table, use the same total for each row, check that the frequencies add up correctly, and make sure the relative frequencies represent the entire dataset.
Once you understand which count belongs in the numerator and which total belongs in the denominator, the method works for raw data, ordinary frequency tables, grouped intervals, and missing-frequency questions.
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