Constructing a Cumulative Frequency Table

Turn a grouped frequency table into a running total, check it ends at the total, and work back from running totals to frequencies.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 30 real past-paper questions below

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What Is Cumulative Frequency?

Cumulative frequency is a running total. You keep adding each frequency to the total so far.

Minutes	Frequency	Cumulative
0 to 10	5	5
10 to 20	11	16
20 to 30	18	34

Key idea

Each cumulative frequency tells you how many values are at or below the end of that class.

Completing the Table

The 1-mark question: complete the cumulative frequency table. Add each new frequency to the one above.

Worked example

1
$f : 5, 11, 18, 16, 7, 3$ — The question: waiting times of 60 people in six classes of 10 minutes.
2
$5$ — The first class has nothing above it, so its running total is just its own frequency, 5.
3
$5 + 11 = 16$ — Add the second frequency to the total so far. 16 people waited up to 20 minutes.
4
$16 + 18 = 34$ — Add the third frequency to the new total, not to 11. Each total builds on the one before.
5
$34 + 16 = 50$ — Same move again: total so far plus the next frequency.
6
$50 + 7 = 57$ — Keep going down the column, one class at a time.
7
$57 + 3 = 60$ — The last total must equal everyone in the question. It is 60, so the adding checks out.

Common mistake

Copying the frequencies into the new column. Each entry must include every class above it.

Key idea

Write every total, even if you are unsure. If one sum is wrong but the totals after it follow on correctly from yours, you can still get a mark.

Reading the Class Labels

The table you fill in often has labels like $0 < t \leq 30$ . That means every value from the start up to 30, not just the class 20 to 30.

If the first class starts at 20, the labels all start at 20: $20 < a \leq 50$ means ages above 20 up to 50.

Worked example

1
$f : 9, 15, 21, 12, 3$ — The question: 60 ages in classes of 10 years, starting at 20.
2
$20 < a \leq 30 : 9$ — The first label covers only the first class, so it is just its frequency.
3
$20 < a \leq 40 : 9 + 15 = 24$ — This label goes from 20 up to 40, so it holds the first two classes.
4
$20 < a \leq 50 : 24 + 21 = 45$ — Up to 50 adds the third class. So 45 people are above 20 and up to 50.
5
$20 < a \leq 60 : 45 + 12 = 57$ — Add the fourth class to the total so far.
6
$20 < a \leq 70 : 57 + 3 = 60$ — The last label covers every class, so it must be the total, 60.

Rule

Check: the last cumulative frequency must equal the total number of values.

Going Back to the Frequencies

Sometimes you have the running totals and need each class's frequency. Subtract the total above.

Worked example

1
$CF : 8, 21, 39, 52, 60$ — The question gives running totals for five classes. You need each class on its own.
2
$f_{1} = 8$ — Nothing comes before the first class, so its frequency is the first running total.
3
$21 - 8 = 13$ — The total of 21 includes the first class's 8. Take it away to leave the second class.
4
$39 - 21 = 18$ — Take away the total above again to leave the third class.
5
$52 - 39 = 13$ — Same move for the fourth class.
6
$60 - 52 = 8$ — And for the last class.
7
$8 + 13 + 18 + 13 + 8 = 60$ — Check: the frequencies add back to the last running total, 60.

Common mistake

Leaving the first class out. Its frequency is the first running total, 8.

Frequencies From a Graph

Some questions give the cumulative frequency graph and ask you to fill in the ordinary frequency table. Read the graph at each class's upper end, then subtract.

Worked example

1
$at 10, 20, 30, 40 : 6, 19, 38, 50$ — A graph for 50 people. Go up from each upper end to the curve, then across to the side axis, and write down the reading.
2
$0 < t \leq 10 : 6$ — The curve starts at 0, so the first class is just the first reading.
3
$10 < t \leq 20 : 19 - 6 = 13$ — The reading at 20 counts everyone up to 20. Take away the 6 up to 10 to leave this class.
4
$20 < t \leq 30 : 38 - 19 = 19$ — Reading at 30 minus the reading at 20.
5
$30 < t \leq 40 : 50 - 38 = 12$ — Reading at 40 minus the reading at 30.
6
$6 + 13 + 19 + 12 = 50$ — Check: the frequencies add up to the total of 50 people.

Common mistake

Writing the readings 6, 19, 38, 50 as the frequencies. Those are running totals, so each class is its reading minus the one before.

Which Method? Read the Question

Look at which column is empty.

You see	You do
Complete the cumulative frequency table	add each frequency to the total so far
Running totals given, find a frequency	subtract the total above
Graph given, complete the frequency table	read at each upper end, subtract the reading before
Draw a cumulative frequency graph	Drawing Cumulative Frequency Diagrams
Median or quartiles	Finding the Median and Quartiles

Key idea

Your last total should match the number of people in the question. If not, check your adding.

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