Cumulative FrequencyCambridge IGCSE Maths: Revision notes

Section 1

What is cumulative frequency?

Cumulative frequency is the running total of frequencies as data values increase. It answers the question 'how many data values are less than or equal to this value?'

To build a cumulative frequency table, add each frequency to the running total of all frequencies before it. The final cumulative frequency total always equals the total number of data values, $n$ .

Key termscumulative frequency

Example

Frequencies 5, 8, 12, 6 for four classes give cumulative frequencies 5, 13, 25, 31.

Section 2

How do I draw a cumulative frequency diagram?

Plot each point at the upper boundary of its class interval against the cumulative frequency up to that point
Join the points with a smooth curve (an 'ogive'), not a straight-line or bar chart
The curve should always be increasing (or level, never decreasing) since cumulative frequency can never go down

Plot points clearly and precisely — mark marks are awarded for correct plotting as well as a smooth, sensible curve.

Key termsogiveupper class boundary

Common mistake

Plotting cumulative frequency against the class midpoint or lower boundary instead of the upper boundary — this is a very common error.

Section 3

How do I estimate the median from a cumulative frequency diagram?

The median is estimated by finding the value of $\frac{n}{2}$ on the cumulative frequency axis, drawing a horizontal line across to the curve, then a vertical line down to the value on the horizontal axis.

This is an estimate because the exact position of data within each class is unknown — the curve only shows totals at class boundaries.

Key termsmedian (from cumulative frequency)

Example

For $n = 80$ data values, read across from cumulative frequency 40 to estimate the median.

Section 4

How do I find quartiles, percentiles and IQR from the diagram?

Using the same method as the median, but at different cumulative frequency positions:

Measure	Cumulative frequency position
Lower quartile ( $Q_1$ )	$\frac{n}{4}$
Median ( $Q_2$ )	$\frac{n}{2}$
Upper quartile ( $Q_3$ )	$\frac{3n}{4}$
$p$ th percentile	$\frac{p}{100} \times n$

The interquartile range is $Q_3 - Q_1$ , and represents the spread of the middle 50% of the data — read both values from the curve, then subtract.

Key termspercentile

Exam tip

Always state clearly which axis value you are reading off — examiners want to see the horizontal line to the curve and the vertical line down, not just a final number.

Must Know

Cumulative frequency is the running total of frequencies; final total equals $n$
Plot points at the UPPER class boundary against cumulative frequency, then join with a smooth curve (ogive)
Median is estimated at cumulative frequency $\frac{n}{2}$
Lower quartile at $\frac{n}{4}$ , upper quartile at $\frac{3n}{4}$ ; IQR $= Q_3 - Q_1$
Percentiles are read at $\frac{p}{100} \times n$
All values read from a cumulative frequency diagram are estimates, not exact

That's the notes covered.

Carry on to the next subtopic.