Cumulative Frequency and Percentiles

Cumulative frequency counts observations at or below each upper class boundary. Plot the lower starting boundary with cumulative frequency zero, then plot each upper class boundary with its running total. Read horizontally from the desired cumulative frequency and vertically down to the data axis.

Read the lower and upper quartiles from cumulative frequencies \(5\) and \(15\), then subtract to estimate \(\operatorname{IQR}=Q_3-Q_1\) in minutes. A reading between class boundaries is an estimate, not an exact observation. The number above a threshold is total frequency minus cumulative frequency at that threshold.

For \(n\) observations, read the lower quartile, median, upper quartile and \(p\)th percentile at cumulative frequencies \(n/4\), \(n/2\), \(3n/4\) and \(pn/100\), respectively. The cumulative-frequency curve must be non-decreasing. Compare groups of different sizes using proportions or percentages rather than raw counts.