Cumulative Frequency Tables and the Ogive
Simple Explanation
A cumulative frequency table runs a "running total" of frequencies up to each class boundary — each row shows how many data values fall AT OR BELOW that point. Plotting these cumulative totals against the upper class boundaries and joining the points produces a smooth, S-shaped curve called an ogive.
Why Do We Need It?
The ogive turns a table of numbers into a visual tool for instantly estimating what proportion of the data falls below any given value — the foundation for reading off medians and quartiles.
See It
An S-shaped cumulative frequency curve rising from 0 to the total frequency as the class boundary increases
Worked Example
Build a cumulative frequency table
Test scores fall into classes: 0-10 (5 students), 10-20 (13), 20-30 (17), 30-40 (10), 40-50 (5). Build the cumulative frequency table.
Why Does This Work?
Since each cumulative total already includes every value below it, plotting these totals against the upper boundary and connecting them traces exactly how the proportion of data "below this point" grows as the boundary increases — always rising (or staying flat), which is why the curve always has this characteristic S-shape.
Real-Life Example
Estimating what percentage of runners finished under a given time
A race organizer wants to quickly estimate what fraction of runners finished within any given time, from grouped finishing-time data.
Plotting an ogive of finishing times lets them read that percentage directly from the graph, for any time value, without recomputing from the raw table.
Practice
Classes 0-5 (3), 5-10 (7), 10-15 (4) have what cumulative frequency at the boundary 10?
MediumCommon mistake
Plotting the cumulative frequency against the LOWER class boundary instead of the UPPER boundary — a cumulative total "up to 20" must be plotted at x=20, not at the class's starting value.
Quick Review
- Cumulative frequency = running total of frequencies up to each class.
- Plot cumulative frequency against the UPPER class boundary.
- Joining the points gives an S-shaped ogive curve.