How Do You Use a Cumulative Frequency Graph to Find the Median and Quartiles?

You find the median and quartiles by using the total cumulative frequency, calculating the positions $\frac{N}{4}$, $\frac{N}{2}$, and $\frac{3N}{4}$, and reading the corresponding values from the horizontal axis. Values read from the graph are estimates.

The Method

A cumulative frequency graph shows how many observations are less than or equal to each value. The final cumulative frequency gives the total number of observations, $N$.

On an IB graph, cumulative frequency is normally plotted against the upper class boundaries. Therefore, the horizontal reading estimates the data value below which the chosen proportion of observations lies. If the curve crosses between grid lines, interpolate using the scale rather than rounding immediately.

For each measure:

1. Calculate its cumulative frequency position.
2. Locate that position on the vertical cumulative frequency axis.
3. Draw or trace a horizontal line to the cumulative frequency curve.
4. From the curve, draw or trace a vertical line down to the horizontal data-value axis.
5. Read the corresponding value.

For example, suppose the graph represents $80$ students. The required positions are

$$Q_1: \frac{80}{4}=20, \qquad Q_2: \frac{80}{2}=40, \qquad Q_3: \frac{3(80)}{4}=60.$$

Therefore, read $Q_1$ at cumulative frequency $20$, the median at $40$, and $Q_3$ at . If the graph gives approximate horizontal-axis values of , , and , then , the median , and .

The interquartile range can then be calculated using

$$\operatorname{IQR}=Q_3-Q_1=39-24=15.$$

A common misconception is to locate quartiles at one-quarter, one-half, and three-quarters of the horizontal axis. This is incorrect: first use these fractions of the total frequency on the vertical axis.

IB Exam Technique

This is SL 4.2: Presentation of data and may be tested in either Paper 1 or Paper 2. Show the quartile positions, read carefully from the graph's scale, and use $\approx$ because graphical readings are estimates. Keep full precision for any later IQR calculation, then round according to the graph's accuracy.