Cumulative frequency is a running total of frequencies, built up to each class's upper boundary. Plotted against the upper boundaries, it forms a smooth S-shaped curve (an ogive) that lets the median, quartiles and percentiles be read off directly, without needing the raw data.

Plot each point at $(\text{upper class boundary}, \text{cumulative frequency})$, plus a starting point at the lower boundary of the first class with cumulative frequency $0$.

Example: times of $80$ runners.

Points to plot: $(40,0),(50,6),(60,22),(70,56),(80,74),(90,80)$, joined with a smooth curve — never straight segments, since an ogive summarises a continuous accumulation.

With $N=80$ total: read across from $y=\frac12N$, $\frac14N$ and $\frac34N$ to the curve, then down to the $x$-axis.

Any percentile is read off using exactly the same "across then down" method, just starting from a different fraction of $N$ on the vertical axis. The $p$th percentile is read from height $y=\dfrac{p}{100}\times N$.

Example: using the same $80$ runners, estimate the $90$th percentile.

Reading across from $y=72$ to the curve and down to the $x$-axis gives approximately $78$ seconds — meaning about $90\%$ of runners completed the race in under $78$ seconds.

Reading up from $x=75$ to the curve gives about $66$ runners taking $75$ seconds or less. Since the curve only ever shows "up to" a value, the number taking longer than $75$ seconds needs a subtraction from the total:

The same subtraction idea extends to finding a range: the number of runners taking between two times is found by reading off the cumulative frequency at each boundary and subtracting one from the other, rather than trying to read a "band" directly off the curve.