Relative frequency tells you what proportion of your data falls into a single category, while cumulative frequency tells you how many data points fall at or below a certain value. They answer fundamentally different questions: relative frequency asks “what share of the total does this category represent?” and cumulative frequency asks “how much of the data has accumulated up to this point?”
How Relative Frequency Works
Relative frequency takes the count for a single category and divides it by the total number of observations. If you surveyed 200 people about their favorite season and 54 said summer, the relative frequency for summer is 54 / 200 = 0.27, or 27%. Each category gets its own relative frequency, and when you add all of them together, they should equal 1.0 (or 100%). In practice, rounding can push the total to something like 1.001, which is normal.
The formula is straightforward: divide the frequency of a category (f) by the total number of data values (n). What makes relative frequency useful is that it converts raw counts into proportions, which lets you compare datasets of different sizes. A class of 30 students and a class of 500 students can be compared side by side once you express their grade distributions as relative frequencies rather than raw counts.
How Cumulative Frequency Works
Cumulative frequency is a running total. You start with the frequency of the first category, then add the second category’s frequency to it, then add the third, and so on. Each entry in a cumulative frequency column represents the total number of observations at or below that point.
Say you tracked how many hours you studied each day of the week: 1.5 on Monday, 2.25 on Tuesday, 2 on Wednesday, 1.75 on Thursday, 1.5 on Friday, 3.25 on Saturday, and 2.5 on Sunday. The cumulative frequency after Monday is 1.5 hours. After Tuesday, it’s 3.75. After Wednesday, 5.75. By Sunday, you reach 15 hours total. That final cumulative value always equals the total of all observations in the dataset.
Cumulative frequency is most useful when your data has a natural order, like test scores, ages, or time intervals. It lets you answer questions like “how many students scored 70 or below?” or “on how many days did fewer than 40 people visit?” In a real example from a 30-day climbing study at Lake Louise, cumulative frequency showed that on 11 of the 30 days, 39 or fewer people climbed the rocks.
Key Differences at a Glance
- What it measures: Relative frequency measures the proportion of data in one category. Cumulative frequency measures the running total of data up to and including a category.
- Output format: Relative frequency is a decimal or percentage (like 0.27 or 27%). Cumulative frequency is a whole number count that grows with each category.
- Final value: All relative frequencies sum to 1.0. The last cumulative frequency equals the total number of observations.
- Best for: Relative frequency is ideal for comparing proportions across groups. Cumulative frequency is ideal for determining how many values fall above or below a threshold.
Cumulative Relative Frequency: The Hybrid
There’s a third measure that combines both concepts. Cumulative relative frequency takes the cumulative frequency at each point and divides it by the total sample size. Instead of telling you that 26 out of 35 cars were priced below $8,341, it tells you that 74% of the cars were priced below that amount. The formula is simply cumulative frequency divided by n (the total number of observations).
This hybrid is especially practical when you want to make percentage-based statements about your data. “What percentage of students scored below 80?” is a cumulative relative frequency question. The final value in a cumulative relative frequency column is always 1.0 (or 100%), since by the last category you’ve accounted for every observation.
How They Look on a Graph
Relative frequency is typically displayed as a relative frequency histogram. It looks identical to a standard bar chart, but the vertical axis shows proportions or percentages instead of raw counts. You can quickly see which categories hold the largest share of your data.
Cumulative frequency uses a different type of graph called an ogive (pronounced “o-jive”). Instead of bars, an ogive plots points at the upper boundary of each class interval and connects them with a line. The line always slopes upward because the running total can only increase or stay flat. To build one, you start at the lowest class boundary with a cumulative frequency of zero, then plot each successive point. If you graph cumulative relative frequency on the vertical axis instead of raw counts, the ogive tops out at 1.0 and lets you read off percentages directly.
When to Use Each One
Use relative frequency when you want to understand the distribution of your data across categories, especially when comparing two datasets of different sizes. If you’re looking at survey results from two cities with different populations, converting to relative frequency puts both on equal footing.
Use cumulative frequency when you care about thresholds. It’s the right tool for questions like “how many orders shipped in under 3 days?” or “what percentage of test-takers scored below the passing mark?” Any time the question involves “at or below” (or “at or above”), cumulative frequency gives you the answer directly without requiring you to add up individual categories by hand.
In practice, frequency tables often include all three columns side by side: frequency, relative frequency, and cumulative frequency. Building the full table takes only a few extra calculations and gives you the flexibility to answer almost any question about how your data is distributed.