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Cumulative Frequency

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Cambridge (CIE) IGCSE — International Maths Extended

Cumulative Frequency

Big idea: Cumulative frequency is just a "running total" of how many data values are less than a certain point — and once you plot that running total as an S-shaped curve, you can read off estimates for the median, quartiles, and percentiles without ever seeing the raw data.

Summary — What This Chapter Covers

Cumulative frequency = a running total of frequencies, added up group by group.
You can build cumulative frequency tables in two styles: step-by-step groups (0–20, 20–40...) or "less than" groups (x<20, x<40...).
A cumulative frequency diagram (ogive) plots cumulative frequency (y-axis) against the upper bound of each class (x-axis) — always producing a stretched S-shape.
You always add one extra point at the start of the data with cumulative frequency = 0.
The diagram lets you estimate the median, lower quartile (Q1), upper quartile (Q3), interquartile range, and any percentile — even though you never saw the individual data values.
Finding these values is a 3-step process: find the position on the y-axis → draw across to the curve → draw down to the x-axis.

1. What Is Cumulative Frequency?

Cumulative literally means "building up." Think of it like a savings account — every time you add a new group's frequency, you add it on top of everything that came before, so the total only ever grows (or stays the same, never shrinks).

Imagine you're timing 50 students on a quiz and you group their times into bands like 25–30 seconds, 30–35 seconds, and so on. The frequency tells you how many students fall into each individual band. The cumulative frequency tells you how many students fall into that band or any band before it — i.e. how many students finished by the end of that band.

The Two Table Styles You'll See

Style A — Frequency table with an extra column. You keep the original group labels (e.g. 20 ≤ x < 40) and add a column where each row = "this row's frequency + everything before it."

Group	Frequency	Cumulative Frequency
0 ≤ x < 20	14	14
20 ≤ x < 40	25	14 + 25 = 39
40 ≤ x < 60	29	39 + 29 = 68
60 ≤ x < 80	12	68 + 12 = 80

Style B — Relabelled "less than" table. Every group is rewritten to start from the very beginning (usually 0), so the label itself tells you the cut-off point: "x < 20," "x < 40," "x < 60"...

Group	Cumulative Frequency
x < 20	14
x < 40	39
x < 60	68
x < 80	80

Handy Trick

If you're given Style B and asked for the original frequency of a group, just subtract consecutive cumulative frequencies. E.g. frequency of "20 ≤ x < 40" = 39 − 14 = 25. This works because each cumulative total already contains everything before it — subtracting removes the overlap.

Core Rule

Cumulative Frequency (row n) = Frequency (row n) + Cumulative Frequency (row n−1)

In plain words: take this group's own frequency, and add it to the running total from every group before it.

Practice Question 1

A shop records the ages of 60 customers in groups: 0≤a<20 (frequency 8), 20≤a<40 (frequency 22), 40≤a<60 (frequency 19), 60≤a<80 (frequency 11). Build the cumulative frequency column.

Practice Question 2

A "less than" cumulative frequency table shows: x<10 → 5, x<20 → 17, x<30 → 34, x<40 → 40. Find the individual frequency of the 20 ≤ x < 30 group.

2. Drawing a Cumulative Frequency Diagram

Once you have your cumulative frequency table, the diagram is just a graph of that running total. The trick that trips most students up is which x-value to plot each point against.

The One Rule That Matters Most

Cumulative frequency is always plotted against the upper bound (end) of the class interval — never the middle, never the start.

Why the upper bound? Take the group 30 ≤ s < 35 with 8 students in it. Those 8 students could have taken anywhere from 30 up to (nearly) 35 seconds — we genuinely don't know their exact times. The only moment we can be certain all 8 of them are "accounted for" is once we reach 35 seconds. That's why the point gets plotted at x = 35, not x = 30 or x = 32.5.

Worked Example — Quiz Times (50 students)

Time (s seconds)	Frequency	Cumulative Frequency
25 ≤ s < 30	3	3
30 ≤ s < 35	8	11
35 ≤ s < 40	17	28
40 ≤ s < 45	12	40
45 ≤ s < 50	7	47
50 ≤ s < 55	3	50

The points to plot are: (30, 3), (35, 11), (40, 28), (45, 40), (50, 47), (55, 50) — plus one extra starting point at (25, 0), because before the lowest possible value (25 seconds), zero students had finished.

Cumulative Frequency 50 | x───x 40 | x───x 30 | x 20 | x 10 | x 0 x──────────────────────────────────────── 25 30 35 40 45 50 55 Time, seconds Notice the "stretched-S" shape: slow start, steep middle, slow finish. It NEVER dips back down — the curve only ever rises or flattens.

Common Mistake

Students often plot against the midpoint of each group (like they do for estimating the mean from grouped data) — that's a completely different technique! For cumulative frequency diagrams, always use the upper bound. Also, don't forget the starting point at (lowest value, 0) — missing it is one of the most common lost marks in exams.

Finally, join the points with a smooth curve (not straight lines) that passes through every plotted point. This takes practice — aim for a graceful S rather than sharp angles.

Practice Question 3

A cumulative frequency table has groups 0≤m<10, 10≤m<20, 20≤m<30, 30≤m<40 with cumulative frequencies 6, 21, 38, 45. What are the (x, y) coordinates of all the points you'd plot, including the starting point?

3. Reading the Diagram: Median, Quartiles & Percentiles

This is the whole point of drawing the diagram in the first place — it lets you estimate statistics you'd otherwise need the raw (ungrouped) data to calculate exactly.

The Golden Method (works for everything in this section)

Step 1: Work out the position you need on the cumulative frequency (y) axis.
Step 2: Draw a horizontal line from that position across until it touches the curve.
Step 3: From that touching point, draw a vertical line straight down to the x-axis and read off the value.

Finding Each Position (for n total data values)

Statistic	Position formula	Meaning
Median	n ÷ 2	The "middle" value
Lower Quartile (Q1)	n ÷ 4	25% of the way through the data
Upper Quartile (Q3)	3n ÷ 4	75% of the way through the data
p-th Percentile	(n × p) ÷ 100	p% of the way through the data

This Is NOT the Same as Finding the Median of a List!

For a raw list of 60 numbers, you'd normally find the median by averaging the 30th and 31st values. But on a cumulative frequency diagram, you simply use n ÷ 2 = 30 directly as your y-axis position. Don't overthink it or add 0.5 — the curve is treated as continuous, so this small distinction from "ordinary" median-finding doesn't apply here.

Worked Example — Phone Call Lengths (n = 100 calls)

Positions needed: median = 100÷2 = 50; lower quartile = 100÷4 = 25; upper quartile = 3×25 = 75.

Reading across from 50, 25, and 75 on the y-axis to the curve, then down to the x-axis, gives:

Statistic	Reading
Lower Quartile	4.2 minutes
Median	6.2 minutes
Upper Quartile	8.2 minutes

From these, the Interquartile Range (IQR) = Upper Quartile − Lower Quartile = 8.2 − 4.2 = 4 minutes. The IQR tells you how spread out the "middle 50%" of the data is — a smaller IQR means the middle chunk of data is tightly bunched together.

Units Reminder

6.2 minutes stays as 6.2 minutes (or "6 minutes 12 seconds") — never write it as "6 m 2 s" or "6 m 20 s"! That's a classic conversion error: 0.2 of a minute = 0.2 × 60 = 12 seconds, not 2 or 20.

Interquartile Range

IQR = Upper Quartile (Q3) − Lower Quartile (Q1)

Measures the spread of the middle 50% of the data — the smaller the IQR, the more consistent/bunched the data.

Estimating "How Many Are Above/Below a Value" — The Reverse Question

Sometimes the question flips the process: instead of giving you a cumulative frequency and asking for the x-value, it gives you an x-value and asks how many data points are above or below it. Just run the golden method backwards:

Step 1: Start on the x-axis at the given value → draw a vertical line up to the curve.
Step 2: From that point, draw a horizontal line across to the y-axis and read the cumulative frequency.
Step 3: That reading = the number of values below your x-value. If asked for the number above, subtract your reading from the total (n).

Worked Example

Using the same phone call diagram (n = 100): the company wants to know how many of the 100 calls lasted more than 12 minutes. Drawing a vertical line up from x = 12 to the curve, then horizontally across, gives a reading of 90.

This means 90 calls lasted less than 12 minutes. Since the question asks for calls longer than 12 minutes: 100 − 90 = 10 calls.

Practice Question 4

A cumulative frequency diagram shows the heights of 80 plants, with the curve reaching a maximum cumulative frequency of 80. What position (on the y-axis) would you use to find (a) the median, (b) the lower quartile, (c) the 90th percentile?

Practice Question 5

On a cumulative frequency diagram for the masses of 120 parcels (max cumulative frequency 120), reading up from 8kg on the x-axis to the curve, then across to the y-axis, gives a value of 96. How many parcels have a mass greater than 8kg?

What to Memorise

Cumulative Frequency

A running total of frequencies, added group by group as you move through the data.

Plotting Rule

Always plot cumulative frequency against the upper bound (end) of each class interval.

Starting Point

Add the point (lowest possible data value, 0) before drawing the curve.

Curve Shape

Smooth "stretched-S." It never comes back down toward the x-axis.

Median Position

n ÷ 2 (not "30th/31st value" like with raw lists).

Lower Quartile Position

n ÷ 4

Upper Quartile Position

3n ÷ 4 (i.e. 3 × n÷4)

p-th Percentile Position

(n × p) ÷ 100

Interquartile Range (IQR)

Upper Quartile − Lower Quartile. Measures spread of the middle 50%.

Reverse Reading

Given an x-value, go up → across → read cumulative frequency (= number below). Subtract from n for "number above."

Concepts Checklist

I can build a cumulative frequency column by running-totalling the frequencies.

I can convert between the "step" table style and the "less than" table style.

I know to plot cumulative frequency against the upper bound of each class, not the midpoint.

I remember to add the starting point (lowest value, 0) before drawing the curve.

I can draw a smooth stretched-S curve through all plotted points.

I can calculate the correct y-axis position for the median, Q1, and Q3.

I can calculate the position for any given percentile.

I can use the "across then down" method to read values off the curve.

I can calculate the interquartile range from Q1 and Q3.

I can reverse the process — starting from an x-value to estimate how many values lie above or below it.

I know not to convert decimal minutes/hours incorrectly (e.g. 6.2 min = 6 min 12 sec, not 6 min 2 sec).

Exam Tips & Common Mistakes

Wrong Axis Value

The #1 lost mark: plotting against the midpoint or lower bound instead of the upper bound of each class. Examiners specifically check your plotted points' x-coordinates.

Show Your Construction Lines

When finding the median/quartiles/percentiles, always draw (and leave visible) your horizontal and vertical construction lines on the graph. Mark schemes often award a method mark just for showing the correct lines, even if your final reading is slightly off due to graph-reading error.

Answers Are Estimates

Because the diagram is built from grouped data, every value you read off (median, quartiles, percentiles) is an estimate — say so if asked to justify your answer, and expect some tolerance in mark schemes (e.g. ± 0.2 of the correct value on the graph).

Forgetting the Starting Point

Missing the (lowest value, 0) point is a very common way to lose a mark on "draw the diagram" questions — the curve needs a beginning as much as an end.

Confusing "Below" and "Above"

When a question gives an x-value and asks "how many are more than this," remember your direct graph reading gives the number less than that value — you must subtract from the total (n) to get "more than."

Straight Lines Instead of a Curve

Joining the points with straight-line segments (like a frequency polygon) instead of a smooth curve can cost marks — cumulative frequency diagrams should always be a smooth stretched-S shape.

Sanity-Check Your Total

Your last cumulative frequency value should always equal the total number of data values (n). If it doesn't, go back and check your addition.

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Cumulative Frequency

Cumulative Frequency

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