IB Maths AA SL Topic 4 — Statistics Toolkit Paper 1 & 2 ~10 min read

Cumulative Frequency Graphs

A cumulative frequency graph is the running total of your data, plotted as a smooth S-shaped curve. They’re how you find quartiles, percentiles, and medians from grouped data — and they’re easy to read once you know which line to draw.

📘 What you need to know

Cumulative frequency = running total of frequencies up to that value.
For grouped data, plot points at the upper boundary of each class with the cumulative frequency.
Join the points with a smooth curve (not straight lines).
To estimate quartiles or percentiles: draw a horizontal line from the y-axis to the curve, then drop a vertical line to the x-axis.
Median = 50% of total. Q₁ = 25%. Q₃ = 75%. p^th percentile = p% of total.
The graph always increases — never goes down.

What is cumulative frequency?

Cumulative frequency is just a fancy word for “running total”. You go through your frequency table and add each frequency to the previous one, building up to the grand total at the end.

Example — building a cumulative frequency row

Take this grouped frequency table for the heights of 30 students:

Height (cm)	Frequency	Cumulative frequency
140 ≤ h < 150	4	4
150 ≤ h < 160	9	13 (4+9)
160 ≤ h < 170	11	24 (13+11)
170 ≤ h < 180	5	29 (24+5)
180 ≤ h < 190	1	30 (29+1)

The cumulative frequency at the top of each class tells you “how many students are this height or shorter“. So 24 students are below 170 cm, 29 are below 180 cm, and so on.

The last cumulative frequency must equal the total number of values (here, 30). If it doesn’t, you’ve added wrong somewhere — check your arithmetic.

How to draw the graph

5 steps to draw a cumulative frequency graph

Build a cumulative frequency column in your table.
Plot each point at the upper boundary of the class, with the cumulative frequency on the y-axis.
Plot a starting point at the lower boundary of the first class with cumulative frequency 0 (the curve must start somewhere).
Join the points with a smooth curve (NOT straight lines!).
Label your axes — the x-axis is the variable (with units), the y-axis is “cumulative frequency”.

Cumulative frequency curve for the 30 students’ heights

🤔 Why do we plot at the upper boundary?

If 4 students are in the class 140 ≤ h < 150, we know all 4 are below 150 cm — but we don’t know exactly where. The safest place to mark “4 students so far” is at the upper boundary of 150 cm. By 160 cm, we’ve now counted 13 students; by 170 cm, 24; and so on.

How to read information from the graph

Cumulative frequency graphs are a two-way street. You can read them in either direction depending on what the question asks.

From x → y

↑→

“How many values are below x?”

Draw a vertical line up from your x-value to the curve, then horizontally to the y-axis. Read off the cumulative frequency.

e.g. “How many students are shorter than 165 cm?”

From y → x

→↓

“What value is the median / quartile / percentile?”

Draw a horizontal line from your y-value to the curve, then vertically down to the x-axis. Read off the value.

e.g. “Find the median height” → start at y = 15 (half of 30).

Finding the median, quartiles, and percentiles

The position on the y-axis depends on the total number of values (n):

What you want	Position on y-axis	For n = 30
Lower quartile (Q₁)	25% of n = 0.25 × n	7.5
Median (Q₂)	50% of n = 0.5 × n	15
Upper quartile (Q₃)	75% of n = 0.75 × n	22.5
p^th percentile	p% of n = (p/100) × n	e.g. 70th = 21

🧠

Memory trick: “y-axis = how many, x-axis = the value”

The y-axis tells you how many students/items. The x-axis tells you the value. So if a question asks “what’s the median” you start on y; if it asks “how many below 165 cm”, start on x.

Finding the IQR

Interquartile range

IQR = Q₃ − Q₁

Just read Q₁ and Q₃ off the graph (using the 25% and 75% horizontal lines), then subtract.

📍

How to estimate the frequency of a class

To find how many values fell in a class like 140 ≤ h < 160, find the cumulative frequency at x = 160 and at x = 140, then subtract. The difference is the frequency of that range.

Worked examples

WE 1

Build a cumulative frequency column

The frequency table shows times taken (in minutes) by 50 students to finish a test. Complete the cumulative frequency column.

Time (min)	Frequency	Cumulative frequency
0 ≤ t < 10	3	?
10 ≤ t < 20	8	?
20 ≤ t < 30	15	?
30 ≤ t < 40	18	?
40 ≤ t < 50	6	?

Add each frequency to the running total. Final cum freq must equal 50. Row 1: 3 → 3 Row 2: 3 + 8 = 11 Row 3: 11 + 15 = 26 Row 4: 26 + 18 = 44 Row 5: 44 + 6 = 50 ✓ Cum freq: 3, 11, 26, 44, 50 always check that the last value matches the total number of data points

WE 2

Read information from a cumulative frequency graph

The graph below shows the lengths in cm, l, of 30 puppies in a training group.

(a) Find an estimate for the median length. (b) Find Q₁ and Q₃, then the IQR. (c) Estimate the percentage of puppies longer than 51 cm.

n = 30 puppiespart (a) — median Median position: 50% × 30 = 15 From y = 15, draw across to curve, drop down: Median ≈ 47 cmpart (b) — quartiles & iqr Q₁ position: 25% × 30 = 7.5 From y = 7.5 → x ≈ 39.5 cm: Q₁ ≈ 39.5 Q₃ position: 75% × 30 = 22.5 From y = 22.5 → x ≈ 51.4 cm: Q₃ ≈ 51.4 IQR = Q₃ − Q₁: 51.4 − 39.5 = 11.9 IQR ≈ 11.9 cmpart (c) — % longer than 51 From x = 51 cm, go up to curve, across: Cumulative frequency at 51 cm ≈ 22. So 22 puppies are ≤ 51 cm. Therefore: Number longer than 51 cm = 30 − 22 = 8 Percentage = 830 × 100 ≈ 26.7% ≈ 26.7% (3 s.f.) “longer than” = total − below; always subtract from n!

WE 3

Estimate the frequency of a class from the curve

Using the puppies graph from WE 2, given the interval 40 ≤ l < 45 was used when collecting data, find the frequency of this class.

Frequency of a class = cum freq at upper boundary − cum freq at lower boundary. Cum freq at l = 45: ≈ 16 Cum freq at l = 40: ≈ 8 Subtract: 16 − 8 = 8 Frequency of class 40 ≤ l < 45 ≈ 8 subtract the two cumulative frequencies — that gives the count in between

WE 4

Find a percentile

Using the puppies graph again (n = 30), find the 70^th percentile of the puppy lengths.

n = 30, 70^th percentile Percentile position = (p/100) × n. Then read off the curve. Position: 70100 × 30 = 21 From y = 21, draw across to curve, drop down to x-axis. Read off: x ≈ 50 cm 70^th percentile ≈ 50 cm 70% of the puppies are 50 cm or shorter

WE 5

Use a cumulative frequency graph to draw a box plot

From the puppies cumulative frequency graph, the smallest value is approximately 35 cm and the largest is 60 cm. Use this to construct a box plot.

From WE 2 we already have all 5 numbers needed. 5-number summary: Min ≈ 35, Q₁ ≈ 39.5, Median ≈ 47, Q₃ ≈ 51.4, Max ≈ 60 Plot the box from Q₁ to Q₃, line at median, whiskers to min/max. Box plot drawn ✓ cumulative frequency graphs are a fast way to get all the numbers needed for a box plot from grouped data

💡 Top tips

Plot at the upper boundary of each class. All values in the class are below this point — that’s why the cumulative frequency belongs there.
Always include a starting point with cum freq 0 at the lower boundary of the first class. The curve has to start somewhere.
Use a smooth curve, not straight lines when joining the points.
Median = 50% of n. Q₁ = 25% of n. Q₃ = 75% of n. p^th percentile = (p/100) × n.
To find the frequency of a class, subtract the cum freq at its upper boundary from its lower.
“Greater than” or “more than” = subtract the cumulative frequency from n. Don’t read it directly.
Always show the lines you draw on the graph (horizontal first, then vertical) — examiners give marks for the method, not just the answer.
The total cumulative frequency at the end must equal n. If it doesn’t, recheck your additions.

⚠ Common mistakes

Plotting at the mid-interval value instead of the upper boundary. Mid-interval is for estimating the mean — for cumulative frequency graphs, use the upper boundary.
Joining points with straight lines. A cumulative frequency graph is always drawn as a smooth curve.
Using n/2, (n+1)/2 instead of 50% × n. For grouped data on a CF graph, just use percentages of n directly. Don’t fuss with odd/even adjustments.
Reading “more than” directly from the graph. The curve gives you “less than or equal to”. To get “more than”, do n − (cum freq).
Forgetting to start the curve at 0. Without a starting point at cum freq 0, the curve looks like it begins mid-air.
Not drawing the read-off lines. Draw a horizontal line, then a vertical one — and leave them in your answer. Examiners look for those.
Misreading the scale. Always check what each gridline is worth before you start reading off values.
Confusing percentage with cumulative frequency. The 70^th percentile is at y = 0.7 × n (a number of values), not at y = 70 directly.

Cumulative frequency graphs are a Paper 2 favourite — every year there’s a 4–6 mark question that boils down to “find the median, IQR, and a percentile”. Master this method and those marks come for free.

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