What is cumulative frequency in the context of Cambridge IGCSE Mathematics?

Cumulative frequency is a statistical concept used to determine the number of observations that fall below a particular value in a data set. In Cambridge IGCSE Mathematics, it helps students understand how data accumulates over a range of values, providing insights into the distribution of data.

How do you construct a cumulative frequency table?

To construct a cumulative frequency table, start by listing the data values in ascending order. Then, create a column for the frequency of each data value. Next, add a cumulative frequency column where each entry is the sum of the frequencies up to and including that row. This helps visualize how the data accumulates across the dataset.

What is the purpose of a cumulative frequency graph in statistics?

A cumulative frequency graph, or ogive, is used to display the cumulative frequency distribution of a dataset. It helps students in Cambridge IGCSE Mathematics to easily identify the median, quartiles, and percentiles, and to understand the overall distribution of the data.

How can cumulative frequency be used to find the median of a dataset?

To find the median using cumulative frequency, first construct a cumulative frequency table. Then, identify the total number of observations and divide by two to find the median position. Locate this position in the cumulative frequency column to determine the median value of the dataset.

What are some common mistakes to avoid when working with cumulative frequency?

Common mistakes include not ordering data correctly before calculating cumulative frequency, miscalculating the cumulative totals, and confusing cumulative frequency with regular frequency. In Cambridge IGCSE Mathematics, it's crucial to carefully follow each step to ensure accurate results.

Why is "Plotting CF against the lower (or middle) class boundary" a common mistake in Cumulative Frequency?

Why it happens: Defaulting to midpoint as for grouped means. How to avoid it: CF is always plotted against the UPPER class boundary. Source: 0580/42 — recurring

Why is "Drawing a bar chart instead of a smooth curve" a common mistake in Cumulative Frequency?

Why it happens: Confusing with a histogram. How to avoid it: CF graph is a SMOOTH CURVE through the plotted points.

Why is "Using $\dfrac{n+1}{2}$ for the median position on a CF curve" a common mistake in Cumulative Frequency?

Why it happens: Mixing the discrete-list rule with the continuous-curve rule. How to avoid it: On a CF curve (continuous data), use n/2, not n+1/2.

Why is "Computing $Q_{1} - Q_{3}$" a common mistake in Cumulative Frequency?

Why it happens: Order slip. How to avoid it: IQR = Q_3 - Q_1 — bigger minus smaller, always positive.

StatisticsCambridge IGCSE Mathematics (0580)

Cumulative Frequency

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Cumulative Frequency, Quartiles and Box Plots — Cambridge IGCSE 0580 Maths Extended (2026)

Build a cumulative frequency table, draw the curve, read off the median, quartiles and IQR. Then construct a box plot. Cambridge wants smooth S-shaped curves and correctly placed box-plot whiskers.

What you’ll learn

Mapped to the Cambridge IGCSE 0580 syllabus (2025-2027).

E11.5 — Construct and interpret cumulative frequency diagrams.
E11.6 — Find the median, quartiles, percentiles, range and interquartile range from a cumulative frequency diagram.
E11.7 — Construct and interpret box-and-whisker plots.

Building a cumulative frequency table

Running total: each row's cumulative is its frequency plus all previous frequencies.

Method. Add a column to the frequency table; cumulative frequency at each row = frequency at that row + cumulative at the previous row.

Worked. Heights of $40$ students:

Height $h$ (cm)	Frequency $f$	Cumulative
$140 < h \le 150$	4	4
$150 < h \le 160$	12	16
$160 < h \le 170$	14	30
$170 < h \le 180$	8	38
$180 < h \le 190$	2	40

The final cumulative equals the total frequency: $40$ .

Plot. Plot cumulative frequency against the UPPER class boundary:

$(150, 4), (160, 16), (170, 30), (180, 38), (190, 40)$ .

Also plot $(140, 0)$ — nothing below the lowest boundary.

Curve. Join with a smooth curve (NOT straight lines between points). The result is an "S-shaped" curve rising from $0$ to the total.

Cumulative = running total of frequencies.
Plot at UPPER class boundary.
Include $(\text{lowest boundary}, 0)$ at the start.
Smooth curve, not straight segments.

Reading off median, quartiles and percentiles

Find the position on the cumulative axis ( $\dfrac{n}{2}, \dfrac{n}{4}, \dfrac{3n}{4}$ ), draw across, then down to the data axis.

Method.

Identify $n$ = total cumulative frequency.
Median position: $\dfrac{n}{2}$ . Quartile positions: $\dfrac{n}{4}$ ( $Q_{1}$ ) and $\dfrac{3n}{4}$ ( $Q_{3}$ ).
From these positions on the cumulative (vertical) axis, draw a horizontal line to the curve, then a vertical line down to the data (horizontal) axis. Read off the value.

Worked. From the heights example ( $n = 40$ ):

Median position: $20$ . Read $20$ on the cumulative axis, follow across to the curve, drop down: median $\approx 163\,\text{cm}$ .
$Q_{1}$ position: $10$ . Read $10$ , drop down: $Q_{1} \approx 156\,\text{cm}$ .
$Q_{3}$ position: $30$ . Read $30$ , drop down: $Q_{3} \approx 170\,\text{cm}$ .
IQR $= 170 - 156 = 14\,\text{cm}$ .

Percentiles. Same method. The $p$ -th percentile is at position $\dfrac{p}{100} \times n$ on the cumulative axis.

Worked. $90$ th percentile of the heights: position $36$ on cumulative axis. Read off: $\approx 178\,\text{cm}$ .

Median: position $\dfrac{n}{2}$ .
$Q_{1}$ : position $\dfrac{n}{4}$ .
$Q_{3}$ : position $\dfrac{3n}{4}$ .
IQR $= Q_{3} - Q_{1}$ .
Percentile: position $\dfrac{p}{100} \times n$ .

Box-and-whisker plots

Five-number summary: min, $Q_{1}$ , median, $Q_{3}$ , max. Box from $Q_{1}$ to $Q_{3}$ , line at median, whiskers to extremes.

A box plot displays five summary statistics: min, $Q_{1}$ , median, $Q_{3}$ , max.

Construction.

Draw a horizontal axis with appropriate scale.
Draw a box from $Q_{1}$ to $Q_{3}$ .
Draw a vertical line inside the box at the median.
Whiskers: lines from the box ends to the min (left) and max (right).
Label clearly with the values.

Worked. Heights data: min $= 142, Q_{1} = 156$ , median $= 163, Q_{3} = 170$ , max $= 188$ .

Box from $156$ to $170$ .
Median line at $163$ .
Whisker from $156$ left to $142$ .
Whisker from $170$ right to $188$ .

Comparing two box plots. Look at:

Median position: who has the higher centre?
IQR (box length): who is more consistent?
Range (whisker-to-whisker): who has more variability overall?

Worked. Two classes' box plots. Class A: median $65$ , IQR $20$ . Class B: median $70$ , IQR $10$ . Comment: "Class B has a higher median ( $70$ vs $65$ ), so on average performed better. Class B also has a smaller IQR ( $10$ vs $20$ ), so its scores are more consistent."

5 numbers: min, $Q_{1}$ , median, $Q_{3}$ , max.
Box from $Q_{1}$ to $Q_{3}$ .
Whiskers to min and max.
Compare two box plots: median + IQR.

Counting from the cumulative curve

How many are $\le$ , $\ge$ , or BETWEEN given values? Read off the cumulative axis.

Cambridge often asks "how many" questions from a cumulative frequency curve.

$\le$ a value. Find the value on the data axis, go UP to the curve, then ACROSS to the cumulative axis. The number you read is "how many are $\le$ that value".

$\ge$ a value. Compute "how many $\le$ " first, then subtract from $n$ : $n(\ge x) = n - n(\le x).$

Between two values. Subtract: $n(a < x \le b) = n(\le b) - n(\le a).$

Worked. From the heights example. How many students are taller than $175\,\text{cm}$ ?

Read at $x = 175$ : cumulative $\approx 34$ (i.e. $34$ are $\le 175$ ).
Number $> 175$ : $40 - 34 = 6$ .

Worked. How many are between $155$ and $170\,\text{cm}$ ?

Read at $170$ : $\approx 30$ .
Read at $155$ : $\approx 10$ .
Between: $30 - 10 = 20$ .

$\le x$ : read off cumulative at $x$ .
$\ge x$ or $> x$ : subtract from total.
Between $a$ and $b$ : cumulative at $b$ minus cumulative at $a$ .

How it’s examined

Cumulative frequency appears every Paper 4 (8-10 marks) — typically a multi-part question: build the table, plot the curve, read off median + quartiles, construct a box plot, compare with another box plot. Examiner reports flag two errors: (i) plotting at midpoint instead of upper boundary, (ii) only commenting on the median when comparing two boxes (forgetting the IQR).

Sources: Cambridge IGCSE Mathematics 0580 syllabus 2025-2027 (E11.5-11.7); 0580/42 Oct/Nov 2024 — Q14 (cumulative frequency + box plot); 0580 Examiner Reports 2022-2024. Last reviewed 2026-05-05.

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Step-by-step worked examples — Cumulative Frequency

Step-by-step solutions to past-paper-style questions on cumulative frequency, written exactly the way a tutor would explain them at the board.

1Build a cumulative frequency table
Extended• cf table
Question
Frequencies: $[0, 10) = 4$ , $[10, 20) = 9$ , $[20, 30) = 12$ , $[30, 40) = 5$ . Make a CF table.
Step-by-step solution
1. Step 1
  CF at upper bound = running sum.
  $\text{CF}: 10 \to 4,\ 20 \to 13,\ 30 \to 25,\ 40 \to 30$
Answer
$\le 10: 4;\ \le 20: 13;\ \le 30: 25;\ \le 40: 30$ .
2Estimate the median from a CF curve
Extended• Adapted from 0580/42 May/Jun 2024 Q11• median
Question
$30$ data points, CF curve given. Estimate the median.
Step-by-step solution
1. Step 1
  Median is at $n/2 = 15$ .
2. Step 2
  Read across from $\text{CF} = 15$ on the y-axis to the curve, then down to the x-axis.
Answer
Read off the curve at CF = 15.
3Estimate quartiles and IQR
Extended• quartiles
Question
Using the same CF curve ( $n = 30$ ), estimate $Q_{1}$ , $Q_{3}$ and the IQR.
Step-by-step solution
1. Step 1
  $Q_{1}$ at $n/4 = 7.5$ . $Q_{3}$ at $3n/4 = 22.5$ .
2. Step 2
  Read across from CF $= 7.5$ and CF $= 22.5$ .
3. Step 3
  $\text{IQR} = Q_{3} - Q_{1}$ .
Answer
Read off both, then subtract for IQR.
4Estimate a percentile
Extended• percentile
Question
Using the same CF curve, estimate the 80th percentile.
Step-by-step solution
1. Step 1
  80th percentile at CF = $0.8 \times 30 = 24$ .
2. Step 2
  Read across from $24$ to the curve, then down.
Answer
Read off at CF = 24.
5Construct a cumulative frequency table from raw data
Core• cf table, construct
Question
Test marks for $40$ students: $[0, 20) = 3$ , $[20, 40) = 7$ , $[40, 60) = 15$ , $[60, 80) = 11$ , $[80, 100) = 4$ . Construct the cumulative frequency table.
Step-by-step solution
1. Step 1
  CF is the running total up to each upper class boundary.
2. Step 2
  Compute each running total.
  $\le 20: 3;\ \le 40: 10;\ \le 60: 25;\ \le 80: 36;\ \le 100: 40$
Answer
$\le 20: 3;\ \le 40: 10;\ \le 60: 25;\ \le 80: 36;\ \le 100: 40$ . Final value must equal $n = 40$ .
6Plot points for a cumulative frequency curve
Core• plot, cf curve
Question
Using the CF table above, state the coordinates of the points you would plot.
Step-by-step solution
1. Step 1
  Plot (upper class boundary, cumulative frequency). Start at $(0, 0)$ .
2. Step 2
  Points: $(0, 0),\ (20, 3),\ (40, 10),\ (60, 25),\ (80, 36),\ (100, 40)$ .
3. Step 3
  Join with a smooth curve.
Answer
$(0, 0), (20, 3), (40, 10), (60, 25), (80, 36), (100, 40)$ — joined with a smooth curve.
7Find the IQR from a CF curve
Extended• iqr
Question
From a CF curve of $40$ test marks, the values read off are $Q_{1} = 38$ and $Q_{3} = 67$ . Find the IQR.
Step-by-step solution
1. Step 1
  $Q_{1}$ at $\dfrac{n}{4} = 10$ on the CF axis; $Q_{3}$ at $\dfrac{3n}{4} = 30$ .
2. Step 2
  $\text{IQR} = Q_{3} - Q_{1}$ .
  $\text{IQR} = 67 - 38 = 29$
Answer
$\text{IQR} = 29$ marks.
8Number of values above a threshold
Extended• Adapted from 0580/42 May/Jun 2024 Q11(c)• count above
Question
From a CF curve for $40$ students, the cumulative frequency at mark $55$ is $22$ . How many students scored more than $55$ ?
Step-by-step solution
1. Step 1
  CF at $55$ = number scoring at most $55$ .
2. Step 2
  Subtract from total $n$ .
  $n(>55) = 40 - 22 = 18$
Answer
$18$ students scored more than $55$ .
Examiner tip
The examiner report flags students who read off the CF at the threshold and write that as the answer. Subtract from $n$ for 'more than' questions.
9Estimate the 90th percentile
Extended• percentile
Question
From a CF curve for $n = 50$ data points, estimate the 90th percentile.
Step-by-step solution
1. Step 1
  Position on the CF axis: $\dfrac{90}{100} \times 50 = 45$ .
2. Step 2
  Read across from CF $= 45$ to the curve, then drop down to the data value.
Answer
Read off the data value at CF $= 45$ (specific number depends on the given curve).
10Compare two distributions using CF curves (Challenge)
Challenge• Adapted from 0580/42 Oct/Nov 2023 Q11• compare, challenge
Question
Two CF curves are drawn for boys' and girls' test scores ( $n = 40$ each). Boys: median $58$ , IQR $20$ . Girls: median $64$ , IQR $14$ . Make TWO comparisons in context.
Step-by-step solution
1. Step 1
  Centre comparison: the girls have a higher median, so on average girls scored higher.
2. Step 2
  Spread comparison: the girls have a smaller IQR, so girls' scores are less spread out / more consistent.
Answer
Girls scored higher on average (median $64$ vs $58$ ) and were more consistent (IQR $14$ vs $20$ ).
Examiner tip
The examiner report consistently flags comparisons that omit context. Always name the variable and the statistic.
11Draw a box plot from a CF curve (Challenge)
Challenge• box plot, challenge
Question
From a CF curve for $n = 40$ data points, the five-number summary reads: min $20$ , $Q_{1} = 38$ , median $52$ , $Q_{3} = 67$ , max $95$ . Describe the box plot you would draw.
Step-by-step solution
1. Step 1
  Box drawn from $Q_{1} = 38$ to $Q_{3} = 67$ , with vertical line at median $52$ .
2. Step 2
  Whiskers extend from $Q_{1}$ down to min $= 20$ and from $Q_{3}$ up to max $= 95$ .
3. Step 3
  All five values are read off the CF curve at CF $= 0, 10, 20, 30, 40$ respectively.
Answer
Box from $38$ to $67$ , median line at $52$ , whiskers to $20$ and $95$ .
12Match a CF curve to a histogram (Challenge)
Challenge• match, histogram, challenge
Question
Sketches of two histograms (P and Q) and two CF curves (X and Y) are given. Histogram P is skewed RIGHT (tall on the left, long tail right); histogram Q is roughly symmetric. CF curve X rises steeply early then flattens; CF curve Y rises gradually, then steeply in the middle, then flattens. Match the histograms to the CF curves.
Step-by-step solution
1. Step 1
  Right-skewed histogram = most data in low classes = CF rises quickly early then flattens. So P ↔ X.
2. Step 2
  Symmetric histogram = data concentrated in middle classes = CF rises slowly at first, steeply in middle, slowly at end (S-shape). So Q ↔ Y.
Answer
P ↔ X (skewed) and Q ↔ Y (symmetric).
Examiner tip
The examiner report flags students who reason backwards. Translate the histogram shape into 'where the data lives', then ask 'where does the CF rise fastest?'.

Key Formulae — Cumulative Frequency

The formulae you need to memorise for cumulative frequency on the Cambridge IGCSE 0580 paper, with every variable defined in plain English and a note on when to use it.

Reading from a CF curve
$\text{Median: } \dfrac{n}{2};\ Q_{1}: \dfrac{n}{4};\ Q_{3}: \dfrac{3n}{4};\ \text{kth pct: } \dfrac{kn}{100}$
$n$
total frequency
When to use
Estimating positional measures from a CF curve.
Interquartile range
$\text{IQR} = Q_{3} - Q_{1}$
When to use
Spread of the middle 50% of data.

Key Definitions and Keywords — Cumulative Frequency

Definitions to memorise and the exact keywords mark schemes credit for cumulative frequency answers — sharpened from recent examiner reports for the 2026 0580 sitting.

Cumulative frequency
Examiner keyword
Running total of frequencies up to and including a given upper class boundary.
Cumulative frequency curve
Examiner keyword
A smooth curve plotting CF against the UPPER class boundary of each class.
Quartile
Examiner keyword
$Q_{1}$ is the 25th percentile; $Q_{3}$ is the 75th. Median = $Q_{2}$ = 50th percentile.
Interquartile range (IQR)
Examiner keyword
$Q_{3} - Q_{1}$ . A measure of spread that ignores extremes.

Common Mistakes and Misconceptions — Cumulative Frequency

The traps other students keep falling into on cumulative frequency questions — taken from recent Cambridge IGCSE 0580 examiner reports and mark schemes — and how to avoid them.

✕Plotting CF against the lower (or middle) class boundary
0580/42 — recurring
Why it happens
Defaulting to midpoint as for grouped means.
How to avoid it
CF is always plotted against the UPPER class boundary.
✕Drawing a bar chart instead of a smooth curve
Why it happens
Confusing with a histogram.
How to avoid it
CF graph is a SMOOTH CURVE through the plotted points.
✕Using $\dfrac{n+1}{2}$ for the median position on a CF curve
Why it happens
Mixing the discrete-list rule with the continuous-curve rule.
How to avoid it
On a CF curve (continuous data), use $\dfrac{n}{2}$ , not $\dfrac{n+1}{2}$ .
✕Computing $Q_{1} - Q_{3}$
Why it happens
Order slip.
How to avoid it
$\text{IQR} = Q_{3} - Q_{1}$ — bigger minus smaller, always positive.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Video lesson

Short walkthrough of the concepts students most often get stuck on.

Cumulative Frequency — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Cumulative Frequency

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Build a cumulative frequency table

2Estimate the median from a CF curve

3Estimate quartiles and IQR

4Estimate a percentile

5Construct a cumulative frequency table from raw data

6Plot points for a cumulative frequency curve

7Find the IQR from a CF curve

8Number of values above a threshold

9Estimate the 90th percentile

10Compare two distributions using CF curves (Challenge)

11Draw a box plot from a CF curve (Challenge)

12Match a CF curve to a histogram (Challenge)

Reading from a CF curve

Interquartile range

Cumulative frequency

Cumulative frequency curve

Quartile

Interquartile range (IQR)

✕Plotting CF against the lower (or middle) class boundary

✕Drawing a bar chart instead of a smooth curve

✕Using n+12\dfrac{n+1}{2}2n+1​ for the median position on a CF curve

✕Computing Q1−Q3Q_{1} - Q_{3}Q1​−Q3​

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Cumulative Frequency — frequently asked questions

✕Using $\dfrac{n+1}{2}$ for the median position on a CF curve

✕Computing $Q_{1} - Q_{3}$