A frequency distribution organises raw data by counting how often each value – or each range of values – occurs. It is the first step in almost every statistical summary: it shows where the data concentrate, how spread out they are, and whether the distribution is symmetric, skewed or has several peaks.
Ungrouped tables list every distinct value and suit discrete data with few distinct values, such as the number of children in a family or dice rolls. Grouped tables sort the values into classes of equal width, which is the only practical option for continuous measurements such as heights, times or prices. Each class here includes its lower limit and excludes its upper limit (the last class also keeps its upper limit), so no value is counted twice.
Alongside the counts, the table shows relative frequency (the share of all observations), its percentage, and cumulative frequency – the running total that tells you how many observations fall at or below each value. Switch the chart to the cumulative view to see the familiar S-shaped ogive.
How to use the Frequency Distribution Calculator
- Data set
- Type of data
- Start of the first class — Lower limit of the first class; must not be above the smallest value.
- Class width — Each class runs from its lower limit up to, but not including, the next one.
- Chart shows
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The data are sorted and either counted value by value or assigned to classes starting at your chosen start value with the chosen width. Relative frequencies divide each count by the total, and cumulative columns add them up from the top. The mean, median, mode and standard deviation are computed from the raw values, not from class midpoints.
Worked example
Twenty households report how many people live in them: 3, 5, 2, 4, 3, 3, 5, 6, 2, 3, 4, 4, 5, 3, 1, 4, 3, 6, 2, 4. Counting gives 1 → 1, 2 → 3, 3 → 6, 4 → 5, 5 → 3 and 6 → 2. Relative frequencies are 5%, 15%, 30%, 25%, 15% and 10%, and the cumulative frequencies 1, 4, 10, 15, 18, 20 – so half the households (10 of 20) have three people or fewer. Grouping into classes of width 2 starting at 0 gives 0–<2: 1, 2–<4: 9, 4–<6: 8 and 6–<8: 2.
How to read a frequency table
- Look for the row with the largest frequency – the mode or modal class.
- Find the row where the cumulative percentage first reaches 50% – that row contains the median.
- Compare the counts above and below the peak to judge skewness.
- Check that the frequencies add up to n and the relative frequencies to 1 (100%).
Formula
Relative frequency = f ÷ n · Cumulative frequency Fᵢ = f₁ + f₂ + … + fᵢ · Grouped class i = [start + (i − 1)·w, start + i·w)
Worked example
Inputs — Data set: 3, 5, 2, 4, 3, 3, 5, 6, 2, 3, 4, 4, 5, 3, 1, 4, 3, 6, 2, 4; Type of data: Ungrouped – one row per distinct value; Chart shows: Frequency distribution.
Results — Frequency table = Value f Rel. f Rel. % Cum. f Cum. %
─────────────────────────────────────────────
1 1 0.05 5% 1 5%
2 3 0.15 15% 4 20%
3 6 0.3 30% 10 50%
4 5 0.25 25% 15 75%
5 3 0.15 15% 18 90%
6 2 0.1 10% 20 100%; Most frequent value or class = 3 – 6 of 20 (30%); Number of observations (n) = 20; Number of rows (values or classes) = 6; Mean = 3.6; Median = 3.5.
Key terms
- Frequency (f)
- The number of observations with a value or in a class.
- Relative frequency
- Frequency divided by the total number of observations.
- Cumulative frequency
- The sum of the frequencies up to and including a value or class.
- Class limits
- The lower and upper ends of a class in a grouped table.
- Ogive
- A line chart of cumulative frequency against upper class boundaries.
Frequently asked questions
What is a frequency distribution?
A table (or chart) showing how many observations take each value or fall in each class.
How do I make a grouped frequency table?
Choose a start at or below the minimum and a class width, list the classes, then count the values in each. With start 0 and width 2 the classes are 0–<2, 2–<4, 4–<6 and so on.
What is cumulative frequency?
The running total of frequencies. If the counts are 1, 3, 6, 5, the cumulative frequencies are 1, 4, 10, 15.
Which class does a value on a boundary belong to?
Here, the class where it is the lower limit: with width 2, the value 4 goes into 4–<6. The largest class also includes its upper limit.
What is the difference between frequency and relative frequency?
Frequency is a count; relative frequency is that count divided by the total, e.g. 6 of 20 = 0.30 or 30%.
How many values can I enter?
Thousands – paste them from a spreadsheet. Ungrouped tables are limited to 200 distinct values; use grouped data beyond that.
Why do the mean and median not use class midpoints?
Because the calculator has the raw data, it uses the exact values. Midpoint estimates are only needed when you have the table but not the data.