Relative cumulative frequency tells you what proportion (or percentage) of your data falls at or below a given value. The formula is straightforward: divide the cumulative frequency for each class by the total number of observations (n). The result is a decimal between 0 and 1, or a percentage between 0% and 100%, that builds upward as you move through your data classes. Here’s how to calculate it step by step.
The Core Formula
Relative cumulative frequency for any class equals:
Relative Cumulative Frequency = Cumulative Frequency ÷ n
Here, “cumulative frequency” is the running total of how many observations fall in that class or any class below it, and “n” is your total sample size. The last class in your table will always have a relative cumulative frequency of 1.00 (or 100%), because by that point you’ve accounted for every observation.
Step-by-Step Calculation
Start with a frequency distribution table. This is a table listing each value or class interval alongside its frequency, which is simply the count of observations in that group. If you don’t have one yet, sort your data from lowest to highest, group it into intervals, and count how many data points land in each interval.
Once your frequency table is ready, follow these steps:
- Step 1: Find the cumulative frequency for each class. For the first class, the cumulative frequency equals its own frequency. For every class after that, add the current class’s frequency to the cumulative frequency of the class before it. You’re building a running total.
- Step 2: Confirm your total. The cumulative frequency of the last class should equal n, your total number of observations. If it doesn’t, recheck your counts.
- Step 3: Divide each cumulative frequency by n. This converts the running count into a proportion. Multiply by 100 if you want a percentage instead.
Worked Example
Suppose you surveyed 40 students about the number of hours they study per week and grouped the results into five class intervals:
- 0–4 hours: frequency = 5
- 5–9 hours: frequency = 12
- 10–14 hours: frequency = 10
- 15–19 hours: frequency = 8
- 20–24 hours: frequency = 5
First, build the cumulative frequencies. The 0–4 class starts at 5. The 5–9 class is 5 + 12 = 17. The 10–14 class is 17 + 10 = 27. The 15–19 class is 27 + 8 = 35. The 20–24 class is 35 + 5 = 40, which matches n.
Now divide each cumulative frequency by 40:
- 0–4 hours: 5 ÷ 40 = 0.125 (12.5%)
- 5–9 hours: 17 ÷ 40 = 0.425 (42.5%)
- 10–14 hours: 27 ÷ 40 = 0.675 (67.5%)
- 15–19 hours: 35 ÷ 40 = 0.875 (87.5%)
- 20–24 hours: 40 ÷ 40 = 1.000 (100%)
Reading the table, you can now say that 42.5% of students study 9 hours or fewer per week, or that 87.5% study 19 hours or fewer. That instant “at or below” reading is the whole point of relative cumulative frequency.
How It Differs From Related Measures
It’s easy to mix up three closely related terms. Plain “frequency” is just a count for a single class. “Cumulative frequency” is the running total of those counts. “Relative frequency” is a single class’s count divided by n, without any accumulation. Relative cumulative frequency combines both ideas: it’s the running total expressed as a proportion of the whole dataset. If you accidentally divide an individual class frequency by n instead of the cumulative frequency, you’ll get relative frequency, not relative cumulative frequency.
Calculating in a Spreadsheet
You can automate the entire process in Excel or Google Sheets. Set up four columns: class interval, frequency, cumulative frequency, and relative cumulative frequency.
In the cumulative frequency column, the first cell simply references the first frequency. Each cell below it adds the current frequency to the cumulative frequency cell above. For example, if your frequencies start in cell B2, put =B2 in C2, then =C2+B3 in C3, and drag that formula down.
For the relative cumulative frequency column, divide each cumulative frequency cell by the total sample size. If your total is in cell C7, write =C2/$C$7 in D2 and drag down. The dollar signs lock the reference to C7 so it doesn’t shift as you copy the formula. Format the column as a percentage if you prefer that display.
Visualizing With an Ogive
The standard graph for relative cumulative frequency is called an ogive (rhymes with “oh jive”). It’s a line graph where the horizontal axis shows data values (specifically the upper endpoints of each class interval) and the vertical axis shows relative cumulative frequency from 0 to 1.
To build one, plot a point at the left endpoint of the very first class with a relative cumulative frequency of zero. Then plot a point at the right endpoint of each class at its corresponding relative cumulative frequency. Connect the points with straight line segments. The resulting curve rises from 0 on the left to 1.0 on the right.
An ogive makes it easy to estimate percentiles. The value where the line crosses 0.25 on the vertical axis is roughly the 25th percentile (first quartile). Where it crosses 0.50 is the median, and 0.75 marks the third quartile. You can read any percentile the same way: find the desired proportion on the vertical axis, trace horizontally to the line, then drop down to the horizontal axis to read the data value.
When You’d Use This
Relative cumulative frequency is especially useful when you want to compare datasets of different sizes. Because the values are expressed as proportions rather than raw counts, a dataset with 40 observations and one with 4,000 observations are directly comparable. It also shows up often in standardized testing, where your score report might say you performed “at or above 85% of test takers.” That statement is a relative cumulative frequency read straight off an ogive or table.
In coursework, you’ll typically encounter it in introductory statistics when building frequency distribution tables. The pattern is always the same: count, accumulate, then divide by n. Once you’ve done it for one dataset, the process applies identically to any other, whether the data is grouped into intervals or listed as individual discrete values.