Cumulative Frequency

• Cumulative frequency = a running total of frequencies, added up group by group.
• You can build cumulative frequency tables in two styles: step-by-step groups (0–20, 20–40...) or "less than" groups (x<20, x<40...).
• A cumulative frequency diagram (ogive) plots cumulative frequency (y-axis) against the upper bound of each class (x-axis) — always producing a stretched S-shape.
• You always add one extra point at the start of the data with cumulative frequency = 0.
• The diagram lets you estimate the median, lower quartile (Q1), upper quartile (Q3), interquartile range, and any percentile — even though you never saw the individual data values.
• Finding these values is a 3-step process: find the position on the y-axis → draw across to the curve → draw down to the x-axis.

Cumulative literally means "building up." Think of it like a savings account — every time you add a new group's frequency, you add it on top of everything that came before, so the total only ever grows (or stays the same, never shrinks).

Imagine you're timing 50 students on a quiz and you group their times into bands like 25–30 seconds, 30–35 seconds, and so on. The frequency tells you how many students fall into each individual band. The cumulative frequency tells you how many students fall into that band or any band before it — i.e. how many students finished by the end of that band.

Style A — Frequency table with an extra column. You keep the original group labels (e.g. 20 ≤ x < 40) and add a column where each row = "this row's frequency + everything before it."

Style B — Relabelled "less than" table. Every group is rewritten to start from the very beginning (usually 0), so the label itself tells you the cut-off point: "x < 20," "x < 40," "x < 60"...