Relative Frequency Formula - Definition, Steps, Examples

Key Points

• The relative frequency formula is $\text{RF} = \frac{f}{n}$, the frequency of a class divided by the total number of observations.

• The NIST e-Handbook of Statistical Methods describes this as the normalised count, being the count in a class divided by the total number of observations.

• Every relative frequency lies between 0 and 1, and the relative frequencies of all classes sum to exactly 1.

• Multiplying by 100 converts a relative frequency to a percentage, which is the usual way of reporting it.

• Relative frequency is the basis of empirical probability, where an observed proportion estimates the chance of an outcome.

What Is the Relative Frequency Formula?

The relative frequency formula turns a raw count into a proportion of the whole:

$$\text{RF} = \frac{f}{n}$$

where $f$ is the frequency of the class and $n$ is the total number of observations. The NIST/SEMATECH e-Handbook of Statistical Methods calls this the normalised count, defined as the count in a class divided by the total number of observations.

Why the Total Is the Denominator

A raw frequency is only interpretable alongside the sample size. Twelve people preferring a product means something quite different in a survey of 20 than in a survey of 2,000.

Dividing by the total removes that dependence and puts every class on a common scale from 0 to 1. Two surveys of different sizes can then be compared class by class.

The proportions must sum to 1 because every observation falls into exactly one class, so the class counts add back to $n$. That gives a free check on any relative frequency table.

Variable Key

Examples of the Relative Frequency Formula

Example 1

In a class of 40 students, 14 scored a B. Find the relative frequency of a B.

$$\text{RF} = \frac{14}{40} = 0.35$$

Final answer: $0.35$, or 35 per cent.

Example 2

A survey has four response categories with frequencies 8, 14, 12 and 6. Find the relative frequency of the first category by dividing by the number of categories.

There are four categories, so dividing by 4 looks natural: $\frac{8}{4} = 2$.

A relative frequency of 2 is impossible, since a proportion cannot exceed 1. The denominator must be the total number of observations, not the number of classes.

$$n = 8+14+12+6 = 40, \qquad \text{RF} = \frac{8}{40} = 0.20$$

Final answer: $0.20$, or 20 per cent.

Example 3

For the same survey, build the full relative frequency table and check it.

$$\frac{8}{40} = 0.20, \quad \frac{14}{40} = 0.35, \quad \frac{12}{40} = 0.30, \quad \frac{6}{40} = 0.15$$

Final answer: $0.20 + 0.35 + 0.30 + 0.15 = 1.00$, so the table is consistent.

Example 4

A die is rolled 60 times and shows a six on 12 rolls. Find the relative frequency and compare it with the theoretical probability.

$$\text{RF} = \frac{12}{60} = 0.20$$

Final answer: $0.20$ against a theoretical $\frac{1}{6} \approx 0.167$.

The observed proportion is an empirical estimate, and it drifts towards the theoretical value as the number of rolls grows.

Example 5

From the survey in Example 3, find the cumulative relative frequency up to the third category.

Add the relative frequencies in order rather than recomputing:

$$0.20 + 0.35 + 0.30 = 0.85$$

Final answer: $0.85$, so 85 per cent of responses fall in the first three categories.

Don't Confuse These Four Frequency Measures

The cumulative versions always end at $n$ and at 1 respectively, which is the quickest way to spot an arithmetic slip in either column.

The Errors That Cost Most Marks

Mistake 1: Dividing by the number of classes

Where it slips in: The class list is the visible structure of the table, so its length gets used as the denominator.

Don't do this: $\frac{f}{\text{number of classes}}$

The correct way: $\frac{f}{n}$, where $n$ is the total of all frequencies. Any relative frequency above 1 is impossible and signals this error immediately.

Students who write the column total at the foot of the frequency column before computing any proportion avoid this entirely, because the denominator is then sitting in front of them.

Mistake 2: Mixing proportions and percentages in one column

Where it slips in: Some entries get multiplied by 100 and others do not, especially when a table is filled in over two sittings.

Don't do this: Record 0.35 and 30 per cent in the same column.

The correct way: Pick one form and apply it to every row. A column that sums to neither 1 nor 100 is the signature of a mixed column.

Mistake 3: Treating relative frequency as theoretical probability

Where it slips in: Both are numbers between 0 and 1 describing the same outcome.

Don't do this: Report an observed proportion of 0.20 as the probability of rolling a six.

The correct way: Keep them distinct. A relative frequency is what happened in this sample, while a theoretical probability is what the model predicts, and they agree only in the long run.

Wrapping Up

• The relative frequency formula is $\text{RF} = \frac{f}{n}$, a class count divided by the total number of observations.

• Dividing by the total, never by the number of classes, is what puts every class on a 0 to 1 scale.

• The relative frequencies of all classes sum to 1, which checks the whole table in one line.

• An observed relative frequency estimates a probability but is not the same thing as a theoretical one.

• To work through data handling with a teacher, explore a 6th grade math tutor or math tuition.

Practice These Three Before Moving On

A bag draw records 15 red, 25 blue and 10 green. Build the relative frequency table, confirm it sums to 1, then give the cumulative relative frequency after blue.

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Read More

• Frequency Formula

• What Is Frequency

• What Is Probability

• What Is a Percent

• Mean Median Mode Formula