# Frequency Table: Definition, How to Make One, Examples

A frequency table is a two-column summary that lists each value or class interval in a dataset next to the number of times it occurs. It turns a disorganized list of numbers into a distribution you can read in seconds. This article covers the definition, the formulas, and a full worked example with class intervals.

## Quick Answer

- A frequency table lists categories or class intervals in one column and the count of observations in each in another [1].
- The frequency column always sums to the sample size $n$ [2].
- Relative frequency is the count divided by $n$, so the relative frequency column sums to 1, up to rounding [1].
- Cumulative frequency adds each row's count to all the rows above it, and the last entry equals $n$ [1].
- Group quantitative data into equal-width, non-overlapping class intervals when individual values are too numerous to list [3][4].

## What Frequency Tables Mean

In plain terms, a frequency table answers one question: how many times did each value or group of values show up? If you poll 20 kindergarteners on their favorite color, the table has one row per color and a count beside it [2].

The precise statistical definition: a frequency is the number of times a value of the data occurs, and a frequency table arranges those values with their frequencies to display the distribution of a variable [1]. A distribution frequency table for quantitative data groups values into intervals, also called bins or classes, and reports the count in each interval [4].

Frequency tables work for both categorical and quantitative data. Categorical data has clearly defined categories, so the table is straightforward. Quantitative data may need grouping, especially when there are many distinct values [2].

## How It Works

Three formulas cover almost everything you will do with a frequency table.

Frequency is a count:

$$f_i = \text{number of observations in category or class } i$$

Relative frequency converts that count into a proportion of the whole:

$$rf_i = \frac{f_i}{n}$$

where $f_i$ is the frequency of row $i$ and $n$ is the total number of observations. Multiply by 100 to get a percentage [5].

Cumulative frequency accumulates the counts down the table:

$$cf_i = f_1 + f_2 + \dots + f_i$$

The first entry of the cumulative relative frequency column equals the first relative frequency, since there is nothing to accumulate yet. The last entry is 1, meaning 100 percent of the data has been accumulated [1][2].

When you group quantitative data, three rules keep the table valid. Class intervals should be mutually exclusive and non-overlapping, each interval should have the same width, and open-ended classes such as "<10" or ">100" should be avoided [3][4].

## Worked Example

Take 20 exam grades from one class and group them into 5 class intervals. Here is the raw data:

| Grade | | | | |
|---|---|---|---|---|
| 45 | 52 | 58 | 61 | 63 |
| 67 | 68 | 71 | 72 | 74 |
| 76 | 78 | 81 | 83 | 85 |
| 88 | 90 | 92 | 95 | 98 |

**Step 1. Count the observations.** There are $n = 20$ grades.

**Step 2. Choose 5 class intervals.** Use 0-59, 60-69, 70-79, 80-89, and 90-100. The three middle intervals are 10 points wide, while the end classes are wider (0-59 covers all low scores and 90-100 includes a perfect score), a common convention for grade bands.

**Step 3. Count the grades in each class.** The 0-59 class holds 45, 52, and 58, so the count is 3. The 60-69 class holds 61, 63, 67, and 68, so the count is 4. The 70-79 class holds 71, 72, 74, 76, and 78, so the count is 5. The 80-89 class holds 81, 83, 85, and 88, so the count is 4. The 90-100 class holds 90, 92, 95, and 98, so the count is 4.

**Step 4. Compute relative frequencies.** Divide each count by 20. For 0-59: $3 / 20 = 0.1500$, or 15.00%. For 60-69: $4 / 20 = 0.2000$, or 20.00%. For 70-79: $5 / 20 = 0.2500$, or 25.00%. For 80-89: $4 / 20 = 0.2000$, or 20.00%. For 90-100: $4 / 20 = 0.2000$, or 20.00%.

**Step 5. Compute cumulative frequencies.** Through 0-59 the cumulative count is 3. Through 60-69 it is 7. Through 70-79 it is 12. Through 80-89 it is 16. Through 90-100 it is 20.

The finished table:

| Class | Frequency | Relative frequency | Relative percentage | Cumulative frequency |
|---|---|---|---|---|
| 0-59 | 3 | 0.15 | 15% | 3 |
| 60-69 | 4 | 0.20 | 20% | 7 |
| 70-79 | 5 | 0.25 | 25% | 12 |
| 80-89 | 4 | 0.20 | 20% | 16 |
| 90-100 | 4 | 0.20 | 20% | 20 |
| **Total** | **20** | **1.00** | **100%** | |

The frequency column sums to 20, and the relative frequency column sums to 1. Both checks pass.

Here is the same grouping in Python with pandas:

```python
import pandas as pd
grades = [45, 52, 58, 61, 63, 67, 68, 71, 72, 74, 76, 78, 81, 83, 85, 88, 90, 92, 95, 98]
bins = [0, 60, 70, 80, 90, 101]
labels = ['0-59','60-69','70-79','80-89','90-100']
df = pd.DataFrame({'grade': grades})
df['class'] = pd.cut(df['grade'], bins=bins, labels=labels, right=False)
freq = df['class'].value_counts().reindex(labels)
rel = (freq / len(df)).round(4)
print(pd.DataFrame({'count': freq, 'rel_freq': rel}))
```

Output:

```
        count  rel_freq
class                  
0-59        3      0.15
60-69       4      0.20
70-79       5      0.25
80-89       4      0.20
90-100      4      0.20
```

## How to Interpret It

Read the frequency column first to see where the mass of the data sits. In the example, the 70-79 class is the most common with 5 grades, and the 0-59 class is the smallest with 3.

Read the relative frequency column when you want to compare groups of different sizes. A count of 5 means little on its own, but 25% of the class tells you the share directly [6]. Relative frequencies are more commonly used for this reason, since they let you compare how often values occur relative to the overall sample size [6].

Read the cumulative frequency column when the question is about thresholds. The cumulative count of 12 through 70-79 means 12 of the 20 students, or 60%, scored below 80. That single number answers a question the raw counts cannot.

If you want a visual version of the same information, each row of the table becomes a bar in a histogram, with the interval on the x-axis and the frequency or relative frequency as the bar height [4]. A frequency polygon connects the midpoints of those bars and makes it easier to compare two distributions side by side [3].

## When to Use It (and when not to)

Use a frequency table when you need a compact summary of one variable's distribution, when you want to spot the most common values or the shape of the data, or when you need relative and cumulative counts for reporting. It is the natural first step before building a histogram or running a chi-squared goodness-of-fit test on a categorical variable [6].

Use grouped classes when the data is quantitative and has too many distinct values to list individually [4]. The choice of bin width is not standardized, but each bin must have the same width [4].

Skip the table when you have very few observations, since a short list is easier to read. Skip it when you need to compare two variables at once, because a single-variable frequency table cannot show that relationship. For pairs of categorical variables, a contingency table is the right structure, and for the two-way counts inside it, see [joint frequency](/blog/data-analysis/joint-frequency-definition-formula).

## Frequency Table vs Frequency Distribution

The two terms overlap, but they are not identical. A frequency distribution is the underlying pattern of how values are spread across categories or intervals. A frequency table is one way to display that distribution in rows and columns [5].

| Feature | Frequency table | Frequency distribution |
|---|---|---|
| What it is | A table of counts by category or class | The pattern of how values spread across categories or classes |
| Form | Rows and columns | Can be a table, graph, or described shape |
| Includes relative frequency | Often, as an extra column | Describes the concept, not a specific layout |
| Typical use | Reading exact counts and shares | Describing the overall shape of the data |

In practice, people use the phrases almost interchangeably. If you want the broader concept first, start with [frequency distribution](/blog/data-analysis/frequency-distribution) and then build the table.

## Common Mistakes

- **Overlapping class intervals.** Intervals like 60-70 and 70-80 double-count the value 70. Fix it by making intervals mutually exclusive, for example 60-69 and 70-79 [3].
- **Unequal class widths.** A 10-point class next to a 20-point class makes the counts incomparable. Fix it by keeping every interval the same width [4].
- **Forgetting the total check.** The frequency column must sum to $n$ and the relative frequency column must sum to 1, up to rounding [1][2]. If they do not, you missed or double-counted an observation.
- **Reporting raw counts when group sizes differ.** A count of 5 out of 20 is not comparable to 5 out of 200. Fix it by reporting relative frequencies [6].
- **Using open-ended classes.** Classes like "<10" or ">100" hide the spread at the tails. Fix it by choosing a closed interval that covers the full range [3].
- **Treating a grouped table as exact.** Once values are binned, the individual values are gone. Keep the raw data if you need exact statistics.

## Limitations

A frequency table summarizes one variable at a time. It cannot show relationships between two variables, and it cannot tell you whether differences between groups are statistically meaningful. It also discards information: once you group values into classes, you no longer know the exact value of any observation, so means and standard deviations computed from a grouped table are approximations.

The table is also sensitive to your bin choices. The same data grouped into 5 classes and into 10 classes can look like different distributions, and a poorly chosen starting point can hide a cluster or create a misleading gap. Always report the class intervals you used so readers can judge the grouping.

## Frequently Asked Questions

### What is a frequency table in statistics?

A frequency table is a summary that lists each value, category, or class interval in a dataset alongside the number of times it occurs [1]. It can also include relative frequency, which is the count divided by the total, and cumulative frequency, which accumulates the counts down the table [1][2].

### How do you make a frequency table from raw data?

List the distinct values or define your class intervals, then count how many observations fall into each one. Add a relative frequency column by dividing each count by the total, and a cumulative column by adding counts as you move down the rows. Check that the frequency column sums to $n$ and the relative column sums to 1 [1][2].

### What are class intervals in a frequency table?

Class intervals are the ranges that grouped quantitative data is divided into, such as 60-69 or 70-79 [4]. They should be mutually exclusive, non-overlapping, and all the same width [3]. Grouping is necessary when the data has too many distinct values to list individually [4].

### What is the difference between frequency and relative frequency?

Frequency is a raw count of how many observations fall in a category or class. Relative frequency is that count divided by the total number of observations, expressed as a fraction, decimal, or percentage [1][5]. Relative frequencies let you compare distributions across samples of different sizes [6].

### Can a frequency table be used for categorical data?

Yes. Categorical data already has clearly defined categories, so building the table is straightforward: one row per category and a count beside it [2]. Frequency tables, pie charts, and bar charts are all standard ways to display the distribution of a single categorical variable [6].

## References

1. [1.3 Frequency, Frequency Tables, and Levels of Measurement - Introductory Statistics 2e | OpenStax](https://openstax.org/books/introductory-statistics-2e/pages/1-3-frequency-frequency-tables-and-levels-of-measurement)
2. [2.1 Introduction to Descriptive Statistics and Frequency Tables - Significant Statistics - beta (extended) version](https://pressbooks.lib.vt.edu/introstatistics/chapter/frequency-frequency-tables-and-levels-of-measurement/)
3. [Frequency distribution - PMC](https://pmc.ncbi.nlm.nih.gov/articles/PMC3117575/)
4. [3.3: Frequency Tables and Histograms - Statistics LibreTexts](https://stats.libretexts.org/Courses/Red_Rocks_Community_College/Introduction_to_Statistics_(RRCC)/03%3A_Visualizing_Data/3.03%3A_Frequency_Tables_and_Histograms)
5. [2.1 - Frequency Tables - Introduction to Statistics and Statistical Thinking](https://open.maricopa.edu/haasstatistics/chapter/2-1-frequency-tables/)
6. [Frequency Tables, Pie Charts, and Bar Charts](https://sites.utexas.edu/sos/guided/descriptive/descriptivec/frequency/)

## Related Articles

- [Frequency Distribution: Definition, Table and Examples](/blog/data-analysis/frequency-distribution)
- [How to Make a Frequency Table in Excel (Step by Step)](/blog/data-analysis/how-to-make-frequency-table-excel)
- [Joint Frequency: Definition, Formula and Examples](/blog/data-analysis/joint-frequency-definition-formula)
- [ANOVA Table Explained: Components, Formulas and Example](/blog/data-analysis/anova-table-explained)
- [What Is a Contingency Table? Definition and Examples](/blog/data-analysis/what-is-a-contingency-table)