Cumulative Frequency and Box Plots
Cumulative frequency turns a grouped frequency table into a graph you can read estimates off. Box plots then summarise those readings into a picture that makes two distributions easy to compare.
1. Building the cumulative frequency table
Cumulative frequency is a running total of the frequencies.
| Time (min) | Frequency | Cumulative frequency |
|---|---|---|
| 0 < t ≤ 10 | 4 | 4 |
| 10 < t ≤ 20 | 7 | 4 + 7 = 11 |
| 20 < t ≤ 30 | 12 | 11 + 12 = 23 |
| 30 < t ≤ 40 | 5 | 23 + 5 = 28 |
| 40 < t ≤ 50 | 2 | 28 + 2 = 30 |
The last cumulative frequency must equal the total. Here it is 30. If it doesn’t, you have an addition error — this check was recommended in lessons and catches the problem immediately.
2. Plotting the graph
Plot the cumulative frequency against the UPPER BOUND of each class.
For the table above, plot (10, 4), (20, 11), (30, 23), (40, 28), (50, 30).
Always use the upper class boundary — the “second number” of the interval. Plotting at midpoints is the classic error, and it shifts every reading you take afterwards. The logic: by the time you reach 20 minutes, 11 values have occurred — the running total is only complete at the top of the class.
Drawing:
Join the points with a SMOOTH FREEHAND CURVE, not with a ruler. Tutors made this point repeatedly. The curve should rise steadily and level off at the total.
- Start at the lower bound of the first class with cumulative frequency 0 — here (0, 0)
- The curve never goes down
A cumulative frequency graph is not a histogram or a bar chart. Drawing a histogram when a cumulative frequency curve was asked for was a recorded error and scores nothing. Check the y-axis: cumulative frequency rises to the total; a histogram shows frequency (or frequency density) per class.
3. Reading the graph
With n values, read across from these values on the y-axis, then down to the x-axis:
| Measure | Read at |
|---|---|
| Lower quartile (Q1) | n/4 |
| Median (Q2) | n/2 |
| Upper quartile (Q3) | 3n/4 |
For n = 30: read at 7.5, 15 and 22.5.
On a cumulative frequency CURVE, use n/2 — not (n+1)/2. The (n+1)/2 rule is for finding the median of a discrete list of values. A cumulative frequency graph estimates from continuous data, where n/2 is correct. Both rules were used in lessons, each correctly, and mixing them up is a real source of confusion. With large n the difference is tiny, but use the right one.
Draw your reading lines on the graph. Horizontal from the y-axis, vertical down to the x-axis — examiners look for them, and they earn method marks.
All values from a graph are ESTIMATES. Mark schemes usually accept a small range, so read as carefully as you can.
Other readings:
- “How many took less than 25 minutes?” — read up from 25 and across: the cumulative frequency
- “How many took more than 25 minutes?” — read that value, then subtract from the total
- “How many took between 15 and 35 minutes?” — read both and subtract
“More than” needs a subtraction. Reading the graph value directly gives you “less than” — misreading this was a documented error.
4. Interquartile range
IQR = Upper quartile − Lower quartile = Q3 − Q1
If Q1 = 17 and Q3 = 31, then IQR = 14.
The IQR measures the spread of the MIDDLE HALF of the data, so it is not affected by extreme values — unlike the range. A smaller IQR means more consistent data.
It is a single number, not an interval. “17 to 31” is not an IQR; 14 is.
Don’t confuse the IQR with the quartiles themselves, or with a box plot — a specific confusion recorded in lessons. The box plot is the picture; the IQR is the width of the box.
5. Box-and-whisker plots
A box plot displays five values:
Minimum, Lower quartile, Median, Upper quartile, Maximum
- The box runs from Q1 to Q3, with a line inside at the median
- The whiskers extend to the minimum and maximum
Reading one: the box width is the IQR; the whisker-to-whisker span is the range.
You need all five values to draw one. Missing the minimum or maximum is the usual omission.
Draw it against a proper scale, aligned with the axis.
Skewness — from the position of the median inside the box:
- Median in the middle → roughly symmetrical
- Median nearer Q1 (left) → positively skewed (tail to the right)
- Median nearer Q3 (right) → negatively skewed
For positive skew, the mean is greater than the median — the long right tail pulls the mean up. Getting this backwards was a specific recorded error, so it’s worth writing down and learning as a fact.
6. Comparing two distributions
This is what the topic is for, and the mark scheme wants two things:
1. Compare an AVERAGE — usually the medians. 2. Compare the SPREAD — usually the IQRs.
And write it in context:
“Group B has a higher median (32 minutes vs 25 minutes), so they generally took longer. Group B also has a smaller interquartile range (9 vs 14), so their times were more consistent.”
Both halves are needed for full marks, and each must refer to the context with the actual figures. A comparison of medians alone typically scores half.
7. Mistakes that cost marks
Plotting at midpoints instead of upper bounds.
Drawing a histogram or bar chart instead of a cumulative frequency curve.
Using a ruler rather than a smooth curve.
Not starting at zero, or not ending at the total.
Using (n+1)/2 on a cumulative frequency curve.
Reading quartiles at the wrong y-values.
Giving “less than” when “more than” was asked — forgetting to subtract.
Writing the IQR as an interval.
Confusing the IQR with the range, or with the quartiles.
Omitting the minimum or maximum from a box plot.
Getting the direction of skew backwards.
Comparing only averages and not spread — or omitting the context.
Frequently asked questions
What is cumulative frequency? A running total of the frequencies.
Where do I plot the points? At the upper bound of each class.
Why the upper bound? Because the running total is only complete at the top of each class.
Should I use a ruler? No — join the points with a smooth freehand curve.
How do I find the median from the graph? Read across at n/2 and down to the x-axis.
Why n/2 and not (n+1)/2? Because the graph estimates from continuous grouped data. The (n+1)/2 rule is for a discrete list.
How do I find the quartiles? Read at n/4 for Q1 and 3n/4 for Q3.
What is the interquartile range? Q3 − Q1 — the spread of the middle 50% of the data.
Why use the IQR instead of the range? It ignores extreme values, so it describes the bulk of the data more reliably.
What five values does a box plot show? Minimum, Q1, median, Q3, maximum.
How do I compare two box plots? Compare an average (medians) and the spread (IQRs), in context with figures.
Quick revision checklist
- I can build a cumulative frequency column as a running total
- I check the final value equals the total
- I plot at upper class bounds
- I start the curve at zero and draw it freehand and smooth
- I can tell a cumulative frequency graph from a histogram
- I read the median at n/2 on a curve
- I know (n+1)/2 is for discrete lists
- I read quartiles at n/4 and 3n/4
- I draw my reading lines on the graph
- I subtract from the total for “more than” questions
- I can calculate the IQR and know it’s a single number
- I know a smaller IQR means more consistent data
- I can draw and read a box plot with all five values
- I can identify skew and know the mean exceeds the median for positive skew
- I compare average and spread, in context
These notes cover cumulative frequency and box plots in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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