How to Draw and Use Cumulative Frequency Graphs
Cumulative frequency is a running total of the frequencies. Once you have it, you plot a cumulative frequency graph and read off the median, the quartiles and the interquartile range. This page shows you how to complete a cumulative frequency table, draw the curve, and use it to estimate the median and spread of a set of data. Worked examples and a practice room with randomised questions are below.The big idea: a running total
Cumulative frequency is a running total of the frequencies. Plotting it gives an S-shaped curve you can read the median and quartiles from.
What cumulative frequency means
The cumulative frequency for a class is the total of that class and every class before it. In other words, it tells you how many values are less than or equal to the upper boundary of that class.
Once the running totals are worked out, you plot each cumulative frequency against the upper boundary of its class and join the points with a smooth curve. The result is used to estimate the median, the quartiles and the interquartile range.
What you read from the curve
The method, step by step
- Make a running total of the frequencies to get the cumulative frequency for each class.
- Plot each cumulative frequency against the upper boundary of its class.
- Join the points with a smooth curve, starting from the lowest boundary at a height of zero.
- Read off the median at ½n, the lower quartile at ¼n and the upper quartile at ¾n.
- Work out the interquartile range as UQ − LQ.
Worked example: step by step
💡 Step 1: complete the cumulative frequency table
The table shows the heights of 40 plants. Complete the cumulative frequency column.
| Height (cm) | Frequency | Cumulative frequency |
|---|---|---|
| 0 to 10 | 4 | 4 |
| 10 to 20 | 11 | 4 + 11 = 15 |
| 20 to 30 | 15 | 15 + 15 = 30 |
| 30 to 40 | 7 | 30 + 7 = 37 |
| 40 to 50 | 3 | 37 + 3 = 40 |
What's happening?
Each cumulative frequency is the running total so far. Add each new frequency to the total before it. The final value (40) must equal the total number of plants, which is a useful check.
💡 Step 2: draw the graph
Using the completed table from Step 1, draw the cumulative frequency graph.
Plot each cumulative frequency against the upper boundary of its class:
\[(10,4),\ (20,15),\ (30,30),\ (40,37),\ (50,40)\]
Start the curve at \((0,0)\) and join the points with a smooth curve.
What's happening?
The point goes at the top of each class because the cumulative frequency counts everything up to that boundary. The curve always rises and is never flat or falling, since a running total can only stay the same or grow.
💡 Step 3: estimate the median
Use the graph to estimate the median height.
The total is \(n = 40\), so read across at \(\tfrac{1}{2}n = 20\):
Median \(\approx 23.3\) cm
What's happening?
Go across from 20 on the cumulative frequency axis to the curve, then straight down to the value axis. Because it is read from a curve, the answer is an estimate.
💡 Step 4: quartiles and the interquartile range
Use the same graph to estimate the lower and upper quartiles and the interquartile range.
Lower quartile at \(\tfrac{1}{4}n = 10\): LQ \(\approx 15.5\) cm
Upper quartile at \(\tfrac{3}{4}n = 30\): UQ \(\approx 30\) cm
\[\text{IQR} = \text{UQ} - \text{LQ} = 30 - 15.5 = 14.5\text{ cm}\]
What's happening?
The quartiles are read the same way as the median, but at one quarter and three quarters of the total. The interquartile range measures the spread of the middle half of the data, ignoring the extremes.
🔑 Key points
- Cumulative frequency is a running total of the frequencies.
- Plot each total against the upper boundary of its class.
- The curve starts at zero and always rises.
- Median at ½n, lower quartile at ¼n, upper quartile at ¾n.
- IQR = upper quartile − lower quartile.
⚠ Common pitfalls
- Plotting against the middle of the class instead of the upper boundary.
- Forgetting to start the curve at zero.
- Reading the median at the wrong height (use ½n, not the largest value).
- Mixing up the quartiles (¼n is lower, ¾n is upper).
- Giving an exact answer: graph readings are estimates.