Statistics: Plotting the Perfect Cumulative Frequency Curve

By Fatima Khan, M.A. Ed·Updated April 18, 2026
A smooth S-shaped cumulative frequency curve showing median and quartile readings.

Where do you plot the points for a cumulative frequency curve?

You MUST plot the cumulative frequency against the UPPER BOUND of each class interval. Do not plot against the midpoint or the lower bound. For example, if the class is 10 < w ≤ 20 and the cumulative frequency is 15, you plot the coordinate (20, 15).

Table of Contents

Statistics questions on Paper 2 are essentially guaranteed marks if you know the mechanics. They don't require sudden leaps of logic; they just require you to follow an algorithm to draw lines on graph paper. This guide from our Ultimate O-Level Mathematics Guide focuses on the mechanical errors that cost students top grades.

1. The Histogram Trap (Frequency Density)

A histogram looks like a bar chart, but the bars are pushed together. In O-Level Math, examiners love giving you tables with unequal class widths (e.g., 0-10, 10-25, 25-50).

If you just plot "Frequency" on the y-axis, you will get 0 marks. You MUST calculate and plot Frequency Density.

Frequency Density = Frequency ÷ Class Width

Add a new row to your table. Calculate the width of each class, divide the frequency by that width, and plot those new numbers on the y-axis instead.

2. Plotting the 'S-Curve'

A Cumulative Frequency (CF) curve should ALWAYS look like a smooth, stretched-out 'S'.

  • Step 1: Create the CF Column. Add up the frequencies as you go down the table. The final number should be the total number of people/things in the study.
  • Step 2: Plot against the UPPER BOUND. The x-coordinate is the highest number in the class range. The y-coordinate is your running CF total.
  • Step 3: Connect smoothly. Do not use a ruler! Connect the dots with a smooth, freehand curve starting from (0,0) or the lowest lower bound.

3. Extracting the IQR (Interquartile Range)

Once your curve is drawn, examiners will ask you to read values off it to calculate the IQR. Let's assume there are 120 students in total (the top of your y-axis).

  • Median (50% mark): 120 ÷ 2 = 60. Go to 60 on the y-axis, draw across to the curve, and down to the x-axis.
  • Lower Quartile (25% mark): 120 ÷ 4 = 30. Go to 30 on the y-axis, across, and down. This is your 'LQ'.
  • Upper Quartile (75% mark): 3 × (120 ÷ 4) = 90. Go to 90 on the y-axis, across, and down. This is your 'UQ'.
IQR = Upper Quartile (UQ) − Lower Quartile (LQ)

Frequently Asked Questions

How do you calculate frequency density for a histogram?▼
Frequency Density = Frequency ÷ Class Width. This accounts for classes of unequal widths on the graph.
How do you find the median from a cumulative frequency curve?▼
Find the half-way point of the total frequency on the y-axis. Draw a line right to the curve, then straight down to read the median off the x-axis.
What is the Interquartile Range (IQR)?▼
It measures the spread of the middle 50% of the data. IQR = Upper Quartile minus Lower Quartile.
Why does my cumulative frequency curve look wrong?▼
You probably plotted against the midpoint or lower bound. You MUST plot the cumulative total against the UPPER bound of each class.

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