# What is the frequency distribution?

Makes a frequency table from a list of numbers, by equal-width classes or by each value, with the frequency, relative frequency, and cumulative frequencies of each row.

- Page: https://www.acalculator.org/statistics/frequency-distribution-calculator
- JSON spec: https://www.acalculator.org/statistics/frequency-distribution-calculator.json
- Version: 22e3257fd34c

## Default answer

Example with the default inputs (Your numbers [59, 62, 66, 68, 71, 74, 75, 77, 78, 81, 83, 83, 84, 85, 86, 88, 90, 92, 95, 97], Group by Classes, Class width 10): The most common row of the frequency distribution is 80 to under 90, with 7 of 20 numbers.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| data | Your numbers | The numbers, separated by commas, spaces, semicolons, or new lines. |
| by | Group by | Equal-width classes, or one row for each different value. |
| width | Class width | How wide each class is. Every class has the same width. |
| start | First class starts at | The lower edge of the first class. Leave it empty to start at a multiple of the width. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| modal | Most common | The class or value with the highest frequency, and that frequency. |
| rows | Rows | How many classes or values the table has. |
| n | Count (n) | How many numbers are in the list. |

## Method

Classes: start = ⌊min ÷ w⌋ × w unless typed; a number x goes to class ⌊(x − start) ÷ w⌋; relative frequency = f ÷ n; cumulative frequency adds f row by row.

## Assumptions

- Classes include their lower edge and leave out their upper edge, so a number on an edge counts in the class that starts there.
- Empty classes between the first and last number stay in the table with a frequency of 0.

## Worked examples

1. data = 59 or 62, by = class, width = 10 gives modal = 80 to under 90, with 7 of 20 numbers, rows = 5, n = 20. Source: OpenStax, Introductory Statistics 2e, §1.3 Frequency, Frequency Tables, and Levels of Measurement (frequency, relative frequency = frequency ÷ total, cumulative relative frequency), https://openstax.org/books/introductory-statistics-2e/pages/1-3-frequency-frequency-tables-and-levels-of-measurement (retrieved 2026-10-05); OpenStax, Introductory Statistics 2e, §2.2 Histograms, Frequency Polygons, and Time Series Graphs (equal-width classes from a starting point), https://openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs (retrieved 2026-10-05).
2. data = 2 or 3, by = value gives modal = 5, with 7 of 20 numbers, rows = 4. Source: OpenStax, Introductory Statistics 2e, §1.3 Frequency, Frequency Tables, and Levels of Measurement (frequency, relative frequency = frequency ÷ total, cumulative relative frequency), https://openstax.org/books/introductory-statistics-2e/pages/1-3-frequency-frequency-tables-and-levels-of-measurement (retrieved 2026-10-05).
3. data = 0.1 or 0.2, by = class, width = 0.1, start = 0.1 gives modal = 0.3 to under 0.4, with 2 of 4 numbers, rows = 3. Source: OpenStax, Introductory Statistics 2e, §2.2 Histograms, Frequency Polygons, and Time Series Graphs (equal-width classes from a starting point), https://openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs (retrieved 2026-10-05).

## FAQ

### What is a frequency distribution?

A frequency distribution is a table that shows how often each value, or each range of values, appears in a data set. Each row gives a class or value and its frequency, the number of times it occurs.

### How do I make a frequency distribution table?

Pick a class width, start the first class at or below the smallest number, and add classes of that width until the largest number is covered. Then count the numbers in each class. For 20 test scores with width 10 starting at 50, the counts are 1, 3, 5, 7 and 4.

### What is relative frequency?

Relative frequency is a row's frequency divided by the total number of values. If 7 of 20 scores are in the 80s, the relative frequency is 7 ÷ 20 = 0.35, or 35%. The relative frequencies add up to 1.

### What is cumulative frequency?

Cumulative frequency adds the frequencies row by row, so each row shows how many values are in it or any row before it. The last row's cumulative frequency is the total count, and its cumulative relative frequency is 1.

### How do I choose the class width?

Divide the range (largest minus smallest) by the number of classes you want, usually 5 to 15, and round up to a convenient number. Wider classes give a simpler table; narrower ones show more detail.

### Which class does a number on a class edge go into?

Each class includes its lower edge and stops just under its upper edge. So with classes 70 to under 80 and 80 to under 90, a score of 80 counts in the 80s.

## Sources

- OpenStax, Introductory Statistics 2e, §1.3 Frequency, Frequency Tables, and Levels of Measurement: frequency, relative frequency (frequency ÷ total) and cumulative relative frequency. https://openstax.org/books/introductory-statistics-2e/pages/1-3-frequency-frequency-tables-and-levels-of-measurement (retrieved 2026-10-05)
- OpenStax, Introductory Statistics 2e, §2.2 Histograms, Frequency Polygons, and Time Series Graphs: classes of equal width from a starting point. https://openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs (retrieved 2026-10-05)