Frequency, Relative Frequency, and Cumulative Frequency

The three columns that turn raw numbers into a table

A frequency distribution table takes a pile of numbers and sorts them into classes, then counts how many land in each one. Most tables show three counts side by side: frequency, relative frequency, and cumulative frequency. Each one answers a different question, and together they give you a full picture of how your data spreads out.

Let's walk through all three with a real example.

The dataset

Say you collected the ages of 20 people. That is your sample, so n = 20. The youngest and oldest are 40 apart, which gives a range of 40. The mean age is 42.8 and the median is 43.

To build the table, you split that range into classes. Sturges' rule points to 6 classes for a sample this size, so each class covers an equal-width slice of the range. That gives you the six age bands you see below.

The full table

Class (age)	Frequency	Relative frequency	Cumulative frequency
23 to 29.67	3	15%	3
29.67 to 36.33	4	20%	7
36.33 to 43	3	15%	10
43 to 49.67	4	20%	14
49.67 to 56.33	3	15%	17
56.33 to 63	3	15%	20

Now let's read each column.

Frequency: the raw count

Frequency is the plain count of how many values fall into a class. Nothing fancy. Look at the first row: 3 people are aged 23 to 29.67. The second class holds 4 people, and so on down the table.

If you add every frequency together, you get back the sample size. Three plus four plus three plus four plus three plus three lands on 20, which is exactly how many people you started with. That check is worth doing every time. If the frequencies don't sum to n, a value got miscounted or dropped.

Frequency alone is what a plain histogram plots. The height of each bar is the count for that class. You can build one from this same data on the histogram maker and see the shape right away.

Relative frequency: the share of the whole

Relative frequency turns each count into a proportion. The formula is simple:

Relative frequency = frequency / total

So the first class, with 3 out of 20, works out to 15%. The second class, 4 out of 20, is 20%. Every row in the table is that same division, written as a percent.

The useful property here is that all the relative frequencies add up to 100 percent. Fifteen plus twenty plus fifteen plus twenty plus fifteen plus fifteen comes to 100. That makes sense, since every person has to fall into exactly one class, and the whole sample accounts for the whole 100 percent.

Relative frequency is what lets you compare two datasets of different sizes. A raw count of 4 means little on its own, but "20 percent of the sample" is something you can hold next to another group. When you want the bars to show shares instead of counts, plot a relative frequency histogram instead of a plain one.

Cumulative frequency: the running total

Cumulative frequency is a running total of the frequency column. You carry the count forward as you go down the classes.

Start at the top with 3. Add the next class of 4 and you are at 7. Add the following 3 and you reach 10. Keep going: 14, then 17, then 20. Each cumulative value tells you how many people fall at or below the top of that class.

That is why the last cumulative value lands on 20, the sample size. By the final class you have accounted for everyone, so the running total has to equal n. If your bottom cumulative number is anything other than the sample size, something upstream is off.

Cumulative frequency is handy for "how many are under X" questions. Reading the table, 10 people are aged 43 or younger, since the median sits at 43 and the cumulative count hits 10 at that boundary. You can pull answers like that straight off the column without recounting anything.

Putting it together

Each column builds on the one before it. Frequency counts. Relative frequency scales that count against the total. Cumulative frequency stacks the counts up. Two quick checks keep you honest: the relative frequencies should total 100 percent, and the last cumulative value should equal the sample size. Both hold in the table above.

You don't have to grind through the arithmetic by hand. Paste your numbers into the frequency distribution table maker and it builds all three columns for you, classes and all. From there you can chart the same data on the histogram maker to see the distribution take shape, and if you are still deciding how many classes to use, the guide on how to choose bins walks through the trade-offs.