Cumulative frequency and percentiles

Cumulative frequency counts observations up to a boundary. Add frequencies successively, plot each total against its upper class boundary, and include zero at the first lower boundary. Draw a smooth increasing curve through the plotted points; cumulative frequency cannot decrease.

Example

For the time classes \(0<t\le10\), \(10<t\le20\), \(20<t\le30\) with frequencies \(4,8,8\), plot \((0,0),(10,4),(20,12),(30,20)\).

For \(N\) observations, read the median at cumulative frequency \(\dfrac N2\), \(Q_1\) at \(\dfrac N4\), \(Q_3\) at \(\dfrac{3N}{4}\), and the \(p\)th percentile at \(\dfrac p{100}N\). Read horizontally to the curve, then vertically down to the data axis.

Here the median is read at \(10\), \(Q_1\) at \(5\), and \(Q_3\) at \(15\). Graph estimates are about \(17.5\), \(11.3\), and \(23.8\) minutes respectively, so IQR is about \(12.5\) minutes. The \(60\)th percentile is \(20\) minutes because \(0.60(20)=12\).

At \(20\) minutes the cumulative frequency is \(12\), so \(20-12=8\) pupils took more than \(20\) minutes. Between two thresholds, subtract their cumulative frequencies. These readings are estimates; use accuracy supported by the plotted curve, and compare percentages when group sizes differ.