Cumulative frequency graphs
- A cumulative frequency graph answers: how many are below this value?
- It is a running total drawn as a rising curve.
- Its best trick is the reverse read: pick a count, go across to the curve, then down to the value.
Running totals, plotted at the UPPER boundary
- Cumulative frequency is the total of everything less than a value.
- Build it by running-summing the class frequencies.
- Plot each total at the upper class boundary. That is the point where the running total is exact.
- Running-sum the class frequencies into cumulative totals.
- Plot each cf against the upper class boundary.
- Start on the axis at the lowest boundary (); end at .
- Join with straight segments (polygon) or a smooth rising curve.
Worked example
Build the table, then a forward read
80 leaves (nearest cm): (8), (20), (38), (10), (4). Find how many are shorter than 3.7 cm.
- Upper boundaries ; cumulative totals .
- 3.7 cm sits in the segment (cf runs 8 → 28).
- Interpolate: leaves (the polygon estimate).
cf is a running total of “how many below”; plot it at the UPPER boundary, start at 0, finish at n.
Worked example
The reverse read: the longest 22 leaves
Same 80 leaves. Estimate the lower boundary of the lengths of the longest 22 leaves.
- “Longest 22” sits above everyone else, so read in at .
- lies on the segment. Go across to the polygon, then down.
- Interpolate: cm.
To find “the largest k”, read IN at cf = n − k, then read OUT to the value. Forward read picks a value; reverse read picks a count.
Polygon vs curve: why the estimates differ
A polygon joins the exact plotted points with straight segments; a smooth curve bends between them. They agree only at the plotted points, so estimates in between differ a little — here the curve gives ~18 leaves under 3.7 cm and a lower boundary of ~6.7 cm, against the polygon's 20 and 6.9. Neither is “right”: both are estimates, since we never know the true shape between points. The polygon is reproducible (everyone draws the same lines), so in the exam it is the safe default unless a curve is asked for.
Reading off counts
The phrasing tells you exactly which read to do. Match the words to the row below before you touch the graph.
| Phrasing | Read-off |
|---|---|
| fewer than V | |
| more than V | |
| between a and b | |
| largest k | |
| smallest k |
Worked example
All four read-offs off one table
cf at is (). Fewer than 20: . More than 30: . Between 20 and 30: . The largest 6 read in at → above 30.
Worked example
Two cf curves on one axes (a genuine exam shape)
Daily commute time for 80 bank staff: cf at is . (a) How many take between 15 and 45 min? (b) On the same axes you also draw a second team's curve — what does the gap between two curves tell you?
- (a) Between two given values is a subtraction: .
- (b) Two curves on one grid let you compare at a glance: the curve further to the right has larger values; where they cross, that many fall below the same time in each group.
Phrase decides the read: fewer than V = cf(V); more than V = n − cf(V); between = cf(b) − cf(a); largest k → read in at cf = n − k.
Common mistake
Examiner note
Your turn— tap to reveal the worked answer (between two values)
cf at is (). Estimate the number between and s (interpolate to each value first).
→ .
Median, quartiles and percentiles off this same curve are the next chapters' job (Chapters 2–3).
Which display for which data (the §1.5 judgement)
The exam sometimes asks you to choose. Pictograms, bar charts and pie charts suit qualitative and ungrouped discrete data. Small discrete sets favour a stem-and-leaf (raw values survive grouping). Grouped continuous data wants a histogram (shape, relative crowding) or a cumulative frequency graph (counts above/below/between, and varied widths are fine). Above all, the display must be clear and not misleading — e.g. a truncated axis that exaggerates a difference loses marks.