Histograms

A class boundary is the exact edge between two classes. If times are recorded to the nearest minute, 1–10 means $0.5\le t<10.5$. The next class, 11–20, starts at 10.5, so there is no gap.

• Nearest 1 unit: move 0.5 outwards.
• Nearest 0.1 unit: move 0.05 outwards.
• Inequalities such as $20\le t<25$ already give exact boundaries.

$\text{frequency density}=\dfrac{\text{frequency}}{\text{class width}}$

1. Find every boundary and class width. Width is upper boundary minus lower boundary.
2. Divide frequency by width. The result is the bar height. Its unit is frequency per horizontal unit, such as competitors per minute.
3. Draw touching bars on linear axes. A zero-frequency class leaves a gap; otherwise adjacent bars touch.
4. Check each area. Class width × frequency density must return the frequency.

For part of one class, use the same fraction of the bar area. This assumes values are spread uniformly inside that class, so the result is an estimate, not an exact count.