A cumulative frequency histogram is a bar chart where each bar’s height represents the total number of data points that fall at or below that bar’s class interval. Unlike a regular histogram, which shows how many values land in each individual group, the cumulative version adds up values as you move from left to right. This makes it easy to answer questions like “how many observations were below a certain threshold?” at a glance.
How It Differs From a Regular Histogram
A standard histogram divides data into equal-width intervals (called classes or bins) and draws a bar for each one. The bar height shows how many data points fall within that single interval. A cumulative frequency histogram uses the same intervals, but each bar’s height includes all values from every previous interval plus the current one.
Consider a dataset of resting pulse rates measured in 63 healthy people. In a regular histogram, you might see that 15 people had a pulse between 75 and 79 beats per minute. In the cumulative version, the bar for that same interval would show 35, because 2 people fell in the 60–64 range, 7 in 65–69, 11 in 70–74, and 15 in 75–79. Add those up and you get 35 total people with a pulse at or below 79.
This running-total property creates a distinctive visual signature: the bars always increase (or stay the same) as you move left to right. They never decrease. The final bar on the far right always equals the total number of observations in the dataset. Mathematically, this is called a monotonic increasing function.
How to Build One Step by Step
Start with a regular frequency table. List your class intervals in order from lowest to highest, and count how many data points fall into each one. Then add a new column for cumulative frequency. For the first row, the cumulative frequency is just the regular frequency. For every row after that, add the current row’s frequency to the cumulative total from the row above.
Here’s what that looks like using the pulse rate data:
- 60–64 bpm: frequency 2, cumulative frequency 2 (3.2% of total)
- 65–69 bpm: frequency 7, cumulative frequency 9 (14.3%)
- 70–74 bpm: frequency 11, cumulative frequency 20 (31.8%)
- 75–79 bpm: frequency 15, cumulative frequency 35 (55.6%)
- 80–84 bpm: frequency 10, cumulative frequency 45 (71.4%)
- 85–89 bpm: frequency 9, cumulative frequency 54 (85.7%)
- 90–94 bpm: frequency 6, cumulative frequency 60 (95.2%)
- 95–99 bpm: frequency 3, cumulative frequency 63 (100%)
Once your table is complete, draw the chart. Place the class intervals along the horizontal axis and the cumulative frequency (or cumulative percentage) on the vertical axis. Draw a bar for each interval with the height matching its cumulative total. The bars should touch each other with no gaps, just like a regular histogram, because the data is continuous.
Labeling and Axis Rules
The vertical axis always shows the cumulative frequency, either as a raw count or as a percentage of the total. The horizontal axis shows the variable you’re measuring. For continuous data like height, weight, or pulse rate, you plot using the endpoints of each class interval. For discrete data (whole-number counts), you use the upper value of each class.
You can also display cumulative percentage instead of raw counts on the vertical axis. This is sometimes called relative cumulative frequency. In the pulse example, saying “55.6% of participants had a resting pulse of 79 or lower” is often more useful than saying “35 out of 63 people,” especially when comparing datasets of different sizes.
Cumulative Histograms vs. Ogives
You’ll sometimes see the term “ogive” (pronounced oh-jive) used alongside cumulative frequency. These are related but not identical. A cumulative frequency histogram uses bars. An ogive is a line graph that plots a point at the upper boundary of each class interval, then connects the dots. Both display the same cumulative data, just in different visual formats.
Ogives tend to be better for reading off specific values, like “at what pulse rate have we captured 50% of the data?” You follow the line to where it crosses the 50% mark on the vertical axis, then drop down to read the corresponding value on the horizontal axis. Cumulative histograms are better for seeing the overall shape and distribution weight at a glance, particularly when you want to compare how quickly data accumulates across intervals.
Why the Shape Always Rises
Because each bar includes everything that came before it, the bars can only get taller or stay the same. No bar can ever be shorter than the one to its left. If you converted the cumulative histogram into percentages and normalized it so the final value equals 1 (or 100%), the result is what statisticians call a cumulative distribution function. This is useful well beyond basic charting. In image processing, for example, cumulative histograms of pixel brightness values drive techniques like histogram equalization, which adjusts contrast by redistributing how pixel values spread across the brightness range.
Practical Uses
Cumulative frequency histograms answer “how much of the data falls below this point?” questions, which come up constantly in real analysis. During a disease outbreak, plotting cumulative case counts by month reveals how quickly the outbreak progressed and what percentage of total cases had occurred by any given date. In quality control, manufacturers use them to check what proportion of products fall within acceptable tolerances. In education, cumulative frequency helps determine percentile rankings on standardized tests.
The pulse rate example illustrates a common health sciences application. If a researcher needed to know what percentage of healthy volunteers had a resting pulse below 80, the cumulative frequency table gives an immediate answer: 55.6%. Without the cumulative approach, you’d need to manually add up four separate frequency counts.
Creating One in Spreadsheet Software
In Excel, the Data Analysis Toolpak has a built-in histogram function. You provide your raw data range and a “bin range” that defines the upper limit of each class interval. Excel generates a frequency table and optional chart. To turn this into a cumulative version, add a column that calculates running totals of the frequency output. Then chart that column as a bar graph.
One formatting detail that trips people up: Excel’s default histogram chart adds gaps between bars. Since histograms represent continuous data, the bars should touch. You can fix this by right-clicking the bars, selecting “Format Data Series,” and setting the gap width to 0%. Google Sheets and other spreadsheet tools follow a similar process, though the menu labels differ slightly.