What Is a Cumulative Frequency Curve?
A cumulative frequency curve, also called an ogive, is a graph that shows how data accumulates across intervals. Instead of showing how many values fall in each class interval (like a histogram), it shows a running total — how many values are less than or equal to each upper class boundary.
Cumulative frequency is a favourite topic of IGCSE Maths examiners because it tests several skills at once: constructing a table, plotting points accurately, drawing a smooth curve, and reading off statistical measures. Questions on this topic regularly appear on Paper 4 (Extended) and can be worth up to 8 marks.
Step 1: Build the Cumulative Frequency Table
You will usually be given a grouped frequency table. Your first job is to add a cumulative frequency column.
Suppose you have the following data about the heights of 80 students:
| Height (cm) | Frequency |
|---|---|
| 140 ≤ h < 150 | 5 |
| 150 ≤ h < 160 | 12 |
| 160 ≤ h < 170 | 25 |
| 170 ≤ h < 180 | 22 |
| 180 ≤ h < 190 | 11 |
| 190 ≤ h < 200 | 5 |
To create the cumulative frequency column, add each frequency to the running total:
| Upper Boundary | Cumulative Frequency |
|---|---|
| 150 | 5 |
| 160 | 5 + 12 = 17 |
| 170 | 17 + 25 = 42 |
| 180 | 42 + 22 = 64 |
| 190 | 64 + 11 = 75 |
| 200 | 75 + 5 = 80 |
Notice that the final cumulative frequency should equal the total number of data values (80 in this case). If it does not, you have made an arithmetic error somewhere.
Step 2: Plot the Points Correctly
This is where many students lose marks. The rules for plotting cumulative frequency points are:
- Plot each point at the upper class boundary, not at the midpoint or lower boundary
- The cumulative frequency goes on the y-axis
- The variable (height, mass, time, etc.) goes on the x-axis
- You may also plot a point at the lowest boundary with a cumulative frequency of 0 to anchor the curve
So for our example, you would plot: (150, 5), (160, 17), (170, 42), (180, 64), (190, 75), (200, 80). You could also start with (140, 0).
The most common mistake is plotting at the midpoint of each class. Remember: cumulative frequency tells you how many values are less than the upper boundary, so the point belongs at that boundary.
Step 3: Draw a Smooth S-Shaped Curve
Join the points with a smooth freehand curve, not straight lines between points. The curve should have the characteristic S-shape (sigmoid shape):
- It starts relatively flat at the bottom left
- Steepens through the middle where most data is accumulating
- Flattens off again at the top right as it approaches the total
Tips for drawing a good curve:
- Use a sharp pencil so the line is thin and precise
- Do not force the curve through every single point — aim for the best smooth fit
- The curve should never go downward — cumulative frequency can only increase or stay the same
- Do not extend the curve beyond your last plotted point unless instructed
Step 4: Reading Off the Median
The median is the middle value. For n data values, read across from the cumulative frequency axis at n/2.
In our example, n = 80, so the median is at the 40th value. Draw a horizontal line from 40 on the y-axis to the curve, then drop down vertically to the x-axis and read off the value. Let us say you read approximately 168 cm.
The median gives you a measure of the central tendency of the data — roughly half the students are shorter than 168 cm and half are taller.
Step 5: Finding the Quartiles and IQR
The quartiles divide the data into four equal parts:
- Lower quartile (Q1) — read across from n/4 = 20 on the cumulative frequency axis
- Upper quartile (Q3) — read across from 3n/4 = 60 on the cumulative frequency axis
- Interquartile range (IQR) = Q3 − Q1
For our data:
- Q1: read at cumulative frequency 20, which might give approximately 162 cm
- Q3: read at cumulative frequency 60, which might give approximately 178 cm
- IQR = 178 − 162 = 16 cm
The interquartile range tells you about the spread of the middle 50 percent of the data. A smaller IQR means the data is more tightly clustered around the median.
Step 6: Finding Percentiles and Other Values
Examiners sometimes ask for specific percentiles or ask how many students scored above or below a certain value.
- To find the 90th percentile: read across from 0.9 × 80 = 72 on the cumulative frequency axis
- To find how many students are taller than 175 cm: read up from 175 on the x-axis to the curve, then across to the y-axis (say you get 56). The number taller than 175 cm is 80 − 56 = 24
Always show your reading lines on the graph. Examiners award marks for the construction lines (horizontal and vertical dashes) even if your final reading is slightly off.
Common Mistakes to Avoid
These errors cost marks year after year:
- Plotting at midpoints instead of upper boundaries — this is the number one error
- Joining points with straight ruler lines — the question usually asks for a smooth curve
- Reading off the wrong axis — make sure you are reading the correct axis for what the question asks
- Forgetting to include reading lines — dashed lines from axis to curve to axis show the examiner your method
- Arithmetic errors in the cumulative frequency table — always check that your final total matches the number stated in the question
- Choosing an awkward scale — make sure your axes use the full grid space available and that each small square represents an easy-to-read value
Box Plots from Cumulative Frequency
Sometimes the question asks you to draw a box-and-whisker plot using the values you read from your cumulative frequency curve. You will need:
- Minimum value (the lowest boundary)
- Q1
- Median
- Q3
- Maximum value (the highest boundary)
Draw the box from Q1 to Q3 with a line at the median, and extend whiskers to the minimum and maximum. Make sure your box plot uses the same scale as your cumulative frequency curve if both appear on the same page.
Why This Topic Is Worth Mastering
Cumulative frequency questions are among the most predictable in IGCSE Maths. The structure is almost always the same: complete a table, draw a curve, read off values. If you practise the method a few times, you can reliably pick up 6 to 8 marks on exam day. That is a significant chunk of your overall grade for a question type that follows a very consistent pattern.
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