What is cumulative frequency and how is it used in Edexcel IGCSE Mathematics?

Cumulative frequency is a statistical tool used to determine the number of observations below a particular value in a data set. In Edexcel IGCSE Mathematics, it helps students understand data distribution and is often used to create cumulative frequency graphs, which can then be used to estimate medians, quartiles, and percentiles.

How do you construct a cumulative frequency table?

To construct a cumulative frequency table, start by listing the data values in ascending order. Then, for each data value, add the frequency of that value to the sum of the frequencies of all previous values. This running total gives you the cumulative frequency for each data point. This table is essential for creating cumulative frequency graphs.

What is the difference between frequency and cumulative frequency?

Frequency refers to the number of times a particular data value appears in a data set. Cumulative frequency, on the other hand, is the sum of the frequencies for all data values up to and including a particular value. While frequency shows the count of individual data points, cumulative frequency provides a running total, helping to analyze the data's distribution.

How can cumulative frequency graphs be used to find the median in a data set?

To find the median using a cumulative frequency graph, locate the point on the vertical axis that corresponds to half the total number of data points. Draw a horizontal line from this point to intersect the cumulative frequency curve, then drop a vertical line from the intersection point to the horizontal axis. The value at this point on the horizontal axis is the median of the data set.

What are some common mistakes to avoid when interpreting cumulative frequency graphs in IGCSE exams?

Common mistakes include misreading the axes, not accurately plotting points, and confusing cumulative frequency with regular frequency. Students should ensure they understand the scale of the graph and correctly identify key points like the median and quartiles. Practicing with past Edexcel IGCSE exam questions can help avoid these errors.

Why is "Plotting cumulative frequency at the midpoint" a common mistake in Cumulative Frequency?

Why it happens: Confused with frequency polygons. How to avoid it: Cumulative frequency: plot at UPPER BOUND of each class. Always. Source: 4MA1 — examiner reports 2022-2024

Why is "Using $(n+1)/2$ for median on a curve" a common mistake in Cumulative Frequency?

Why it happens: Mixing rules for raw vs grouped data. How to avoid it: On a cumulative frequency curve: n/2. The +1 is for raw lists only.

Why is "Confusing IQR with range" a common mistake in Cumulative Frequency?

Why it happens: Both are spread. How to avoid it: Range = max − min (full spread). IQR = Q_3 - Q_1 (middle 50% only).

Why is "Drawing straight lines instead of a smooth curve" a common mistake in Cumulative Frequency?

Why it happens: Joining points like a polygon. How to avoid it: Cumulative frequency CURVE — smooth S-shape (ogive). Connect with curves, not straight lines.

Statistics and ProbabilityEdexcel IGCSE Mathematics (4MA1)

Cumulative Frequency

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Cumulative Frequency Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Cumulative frequency curves, quartiles, IQR, box plots. The S-curve and box-plot comparisons are tested every series.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

6.4 — Construct a cumulative frequency table from grouped data.
6.4 — Plot and interpret a cumulative frequency curve.
6.4 — Find the median and quartiles from a cumulative frequency curve.
6.4 — Calculate and interpret the interquartile range.
6.4 — Construct and compare box plots.

Cumulative frequency tables

Running total. Plot at the UPPER BOUND.

Cumulative frequency is a running total — the count of values up to and including a given point.

Worked example. Times: $0$ – $10$ (5), $10$ – $20$ (12), $20$ – $30$ (8), $30$ – $40$ (5). Total $n = 30$ .

Cumulative table:

Time $\le t$	Cumulative freq
$\le 10$	5
$\le 20$	17
$\le 30$	25
$\le 40$	30

KEY RULE. Plot cumulative frequency at the UPPER BOUND of each class — NOT the midpoint, NOT the lower bound.

Why? Because "cumulative frequency at $t = 20$ " means "how many had time UP TO 20". The class $10$ – $20$ contributes its full 12 to that count.

Edexcel tip. Always start the curve at $(0, 0)$ if there are no values below the first class — otherwise the curve doesn't pass through the origin.

Running total.
Plot at upper bound.
Start curve at zero where appropriate.
Final cumulative = total $n$ .

The cumulative frequency curve

Smooth S-shape (ogive). Read off median and quartiles.

The curve. Plot the cumulative frequencies at the upper bounds, then connect with a SMOOTH CURVE (not straight lines). The shape is an "S" or "ogive".

Median from the curve. Median position is $n/2$ .

Worked example. $n = 60$ . Median position = $30$ .

Find $30$ on the y-axis.
Read across to the curve.
Read down to the x-axis.
That x-value is the median.

Quartiles from the curve.

Quartile	Position
$Q_1$ (lower quartile)	$n/4$
Median ( $Q_2$ )	$n/2$
$Q_3$ (upper quartile)	$3n/4$

Worked example. $n = 60$ . $Q_1$ at $y = 15$ ; median at $y = 30$ ; $Q_3$ at $y = 45$ .

Interquartile range (IQR). $\text{IQR} = Q_3 - Q_1$ . Measures the spread of the middle 50% — robust to outliers.

Worked example. $Q_1 = 18$ , $Q_3 = 32$ . IQR $= 14$ .

Edexcel tip. Mark schemes give ±1 unit tolerance for graph readings (because precision depends on graph scale). Always show your construction lines on the graph — examiners credit the method.

Connect points with a smooth curve.
Median at $y = n/2$ .
$Q_1$ at $y = n/4$ ; $Q_3$ at $y = 3n/4$ .
IQR = $Q_3 - Q_1$ .
Show construction lines.

See the full worked example for cumulative frequency →

Box plots

Five-number summary as a picture.

A box plot displays five numbers:

Minimum
Lower quartile $Q_1$
Median
Upper quartile $Q_3$
Maximum

Construction.

Box from $Q_1$ to $Q_3$ with a line at the median.
Whiskers from min to $Q_1$ and from $Q_3$ to max.

Worked example. $\min = 5$ , $Q_1 = 12$ , median $= 18$ , $Q_3 = 24$ , $\max = 40$ .

Box: 12 to 24.
Median line at 18.
Left whisker: 5 to 12.
Right whisker: 24 to 40.

Skewness. Compare the two whisker lengths and the position of the median in the box.

Right whisker much longer → POSITIVE skew (long tail to the right).
Left whisker much longer → NEGATIVE skew.
Symmetric: median in middle of box, whiskers similar length.

Comparing box plots. Edexcel asks "compare the two distributions". TWO statements needed:

One about CENTRE (compare medians).
One about SPREAD (compare IQRs).

Worked example. Group A: median 12, IQR 6. Group B: median 18, IQR 4.

"Group B has a higher median (18 vs 12), so on average Group B values are larger."
"Group A has a larger IQR (6 vs 4), so Group A values are more spread out."

Edexcel tip. Always state comparisons IN CONTEXT — say "students" or "marks", not just numbers.

Five-number summary.
Box: $Q_1$ to $Q_3$ with median.
Whiskers: min to $Q_1$ , $Q_3$ to max.
Compare medians AND IQRs in context.

See the full worked example for cumulative frequency →

How it’s examined

Cumulative frequency + box plots is a 6-8 mark question on Paper 2H every series. Compare-two-distributions is the most common follow-up. Examiner reports flag: (1) plotting at midpoints, (2) drawing straight lines instead of a curve, (3) comparison statements without context.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/2H May/Jun 2024, 4MA1/1H Jan 2024. Last reviewed 2026-05-07.

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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
Foundation• cumulative frequency
Question
Times: $0 \le t < 10$ (5), $10 \le t < 20$ (12), $20 \le t < 30$ (8), $30 \le t < 40$ (5). Construct cumulative frequency.
Step-by-step solution
1. Step 1
  Cumulative = running total of frequencies.
2. Step 2
  $5, 5+12=17, 17+8=25, 25+5=30$ .
Answer
$5, 17, 25, 30$
Examiner tip
Plot cumulative frequency at the UPPER BOUND of each class.
2Median from cumulative frequency curve
Higher• Adapted from 4MA1/2H May/Jun 2024 Q17• median
Question
60 students. Read the median from the curve.
Step-by-step solution
1. Step 1
  Median position: $n/2 = 30$ on the y-axis.
2. Step 2
  Read across to the curve, then down to the x-axis. The x-value is the median.
Answer
Median = (whatever the x-value is at $y = 30$ ).
Examiner tip
Use $n/2$ for cumulative frequency curves (not $(n+1)/2$ ). Mark schemes accept ±1 unit tolerance for graph readings.
3Interquartile range
Higher• IQR
Question
$n = 60$ . Lower quartile from $y = 15$ , upper quartile from $y = 45$ . If $Q_1 = 18$ and $Q_3 = 32$ , find IQR.
Step-by-step solution
1. Step 1
  IQR = $Q_3 - Q_1$ .
2. Step 2
  $32 - 18 = 14$ .
Answer
IQR = $14$
4Box plot from a five-number summary
Higher• Adapted from 4MA1/1H Jan 2024 Q21• box plot
Question
Min = $5$ , $Q_1 = 12$ , median = $18$ , $Q_3 = 24$ , max = $40$ . Describe the box plot.
Step-by-step solution
1. Step 1
  Whiskers from 5 to 12 and 24 to 40. Box from 12 to 24, median at 18.
2. Step 2
  Note: the right whisker is much longer — distribution skewed right (positive skew).
Answer
Box: 12–24 with median at 18. Left whisker: 5–12. Right whisker: 24–40.
Examiner tip
Box plot summarises 5 numbers: min, $Q_1$ , median, $Q_3$ , max. Compare two box plots: comment on median (centre) AND IQR (spread).

Key Formulae — Cumulative Frequency

The formulae you need to memorise for cumulative frequency on the Pearson Edexcel IGCSE 4MA1 paper, with every variable defined in plain English and a note on when to use it.

Quartile positions (cumulative frequency curve)
$Q_1 \to n/4, \quad \text{median} \to n/2, \quad Q_3 \to 3n/4$
$n$
total frequency
When to use
Reading quartiles from a cumulative frequency curve.
Example
$n = 80$ : $Q_1$ at $y = 20$ ; median at $y = 40$ ; $Q_3$ at $y = 60$ .
Interquartile range
$\text{IQR} = Q_3 - Q_1$
When to use
Measuring spread of middle 50%.

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 Pearson Edexcel IGCSE 4MA1 sitting.

Cumulative frequency
Examiner keyword
Running total of frequencies — count of values up to and including a given point.
Median (from curve)
The value at cumulative frequency $n/2$ on the curve.
Quartiles
Examiner keyword
$Q_1$ at $n/4$ , $Q_3$ at $3n/4$ . They divide data into quarters.
Interquartile range (IQR)
Examiner keyword
Spread of the middle 50%: $Q_3 - Q_1$ . Robust to outliers.
Box plot
Examiner keyword
Visual summary of min, $Q_1$ , median, $Q_3$ , max.

Common Mistakes and Misconceptions — Cumulative Frequency

The traps other students keep falling into on cumulative frequency questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Plotting cumulative frequency at the midpoint
4MA1 — examiner reports 2022-2024
Why it happens
Confused with frequency polygons.
How to avoid it
Cumulative frequency: plot at UPPER BOUND of each class. Always.
✕Using $(n+1)/2$ for median on a curve
Why it happens
Mixing rules for raw vs grouped data.
How to avoid it
On a cumulative frequency curve: $n/2$ . The +1 is for raw lists only.
✕Confusing IQR with range
Why it happens
Both are spread.
How to avoid it
Range = max − min (full spread). IQR = $Q_3 - Q_1$ (middle 50% only).
✕Drawing straight lines instead of a smooth curve
Why it happens
Joining points like a polygon.
How to avoid it
Cumulative frequency CURVE — smooth S-shape (ogive). Connect with curves, not straight lines.

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

2Median from cumulative frequency curve

3Interquartile range

4Box plot from a five-number summary

Quartile positions (cumulative frequency curve)

Interquartile range

Cumulative frequency

Median (from curve)

Quartiles

Interquartile range (IQR)

Box plot

✕Plotting cumulative frequency at the midpoint

✕Using (n+1)/2(n+1)/2(n+1)/2 for median on a curve

✕Confusing IQR with range

✕Drawing straight lines instead of a smooth curve

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Cumulative Frequency — frequently asked questions

✕Using $(n+1)/2$ for median on a curve