The idea, step by step
Cumulative means running total
A cumulative frequency counts observations up to a boundary. Add each class frequency to the total so far. The final cumulative frequency must equal the sample size, and the running totals cannot decrease.
For an individual ordered data set, this lesson uses the nearest-rank percentile: for 0<P≤100, the Pth percentile is at position ceil(Pn/100). This is a stated convention, not the only percentile rule used by software.
Read a grouped percentile carefully
On a cumulative-frequency graph, find the desired fraction of the total on the vertical scale, move across to the graph, then down to the measurement scale. If a target lies inside a class, a straight-line estimate assumes data are spread evenly through that class.
An estimated percentile is not a prediction for an individual person. For example, a 90th percentile of 40 minutes means roughly 90% of the observations are at or below 40 minutes, not that everyone takes 40 minutes.
A common trap
The frequency of a class equals its cumulative frequency.
Instead: Subtract consecutive cumulative totals to recover the class frequency.
See the thinking
A nearest-rank percentile
The ordered scores are 10, 12, 14, 16, 18, 20, 22, 24, 26, 28. Find P70 using nearest rank.
- n=10, so the rank isceil(0.70×10)=7.
- The seventh value is 22.
Estimate from a cumulative graph
Estimate the median completion time for the 40 times shown.
- Half of 40 is 20, so locate cumulative frequency 20.
- The graph rises from 12 at 20 minutes to 24 at 30 minutes.
- 20 is 8 of the 12 observations into that interval:20+(8/12)×10≈26.7.
Check your understanding
Cumulative totals are 5, 13, 20. How many observations are in the second class?
Show answer
8
13−5=8.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Frequencies are 3, 7, 4, 6. Write the cumulative frequencies.2 marks
- Cumulative frequencies are 4, 11, 19, 25. Find the original frequencies.2 marks
- For ordered values 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, find P80 by nearest rank.2 marks
- A sample has 60 values. Which rank gives P25 under nearest rank?2 marks
- Use the graph to estimate P80 for 40 times.2 marks
- A graph puts P90 at 45 minutes. Explain this in context.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Frequencies are 3, 7, 4, 6. Write the cumulative frequencies.2 marks
Start with 3.
Add each next frequency to the running total.
3, 10, 14, 20. - Cumulative frequencies are 4, 11, 19, 25. Find the original frequencies.2 marks
First frequency 4.
Successive differences:11−4, 19−11, 25−19.
4, 7, 8, 6. - For ordered values 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, find P80 by nearest rank.2 marks
ceil(0.8×10)=8.
The eighth value is 16.
16 - A sample has 60 values. Which rank gives P25 under nearest rank?2 marks
ceil(0.25×60)=15.
15th - Use the graph to estimate P80 for 40 times.2 marks
0.80×40=32.
Cumulative 32 corresponds to 40 minutes.
40 minutes - A graph puts P90 at 45 minutes. Explain this in context.2 marks
A percentile locates an ordered position.
It does not give the probability that every future task finishes by 45 minutes.
About 90% of recorded completion times are 45 minutes or less.
Curriculum reference & lesson scope
Australian Curriculum v 9 and SA Curriculum Mathematics
Year 10 continuous distributions, bivariate data, two-way categorical tables, bias and investigations, with the approved percentile and standard-deviation content included in the normal chapter.
Official curriculum source ↗