Statistics with Ungrouped Data

Lessons

Senior 1 · Mathematics

Statistics with Ungrouped Data

5 lessons

Lesson 2 of 5

Frequency tables and cumulative frequency

22 min

Thirty numbers in a notebook

For a health programme, a village leader wrote down the number of people living in each of the 30 households of her umudugudu, in the order she visited them:

3, 7, 2, 4, 4, 6, 5, 5, 4, 3, 4, 4, 7, 5, 2, 3, 5, 6, 3, 3, 6, 2, 5, 5, 4, 8, 6, 4, 4, 5

As a list, the numbers say very little. How many households have exactly four people? What size is most common? A frequency table answers such questions at a glance.

Definition

Frequency distribution table (frequency table): a table that lists each value of a variable together with its frequency, the number of times it occurs.

Notation: the values are called $x$ and their frequencies $f$ . The total of the frequencies is written $\sum f$ ("sigma f", the sum of the frequencies), and it equals $n$ , the number of data values.

Building a frequency table

Find the smallest and the largest value (here 2 and 8), and list every value in between in the first column.
Go through the raw data once, in order. For each value, draw a tally mark in its row and tick the value off the list.
Count the tally marks in each row to get the frequency.
Check that $\sum f = n$ .

The frequencies add up to 30, the number of households. That check catches a missed or double-counted value.

Definition

Cumulative frequency of a value: the sum of the frequencies of that value and all the values below it. It is a running total, and it counts how many data values are less than or equal to that value.

Example: the cumulative frequency of 4 people is $3 + 5 + 8 = 16$ : sixteen households have 4 people or fewer.

Each cumulative frequency is the one above it plus the new frequency: $3,\ 3 + 5 = 8,\ 8 + 8 = 16,\ \ldots,\ 29 + 1 = 30$ .

People in household $(x)$	Frequency $(f)$	Cumulative frequency	Meaning of the cumulative frequency
2	3	3	3 households have 2 people or fewer
3	5	8	8 households have 3 people or fewer
4	8	16	16 households have 4 people or fewer
5	7	23	23 households have 5 people or fewer
6	4	27	27 households have 6 people or fewer
7	2	29	29 households have 7 people or fewer
8	1	30	30 households have 8 people or fewer

Important

Two checks on every cumulative frequency column

It never goes down, because each step adds a frequency, and frequencies are never negative.
Its last entry equals $\sum f = n$ , because by the largest value every data value has been counted.

And the key idea for using it: the number of values greater than $x$ is $n$ minus the cumulative frequency of $x$ .

Worked example 1: reading a cumulative frequency table

Use the household table above. How many households have (a) at most 4 people, (b) more than 5 people, (c) from 3 to 6 people? (d) What percentage of households have at most 4 people?

Solution

"At most 4" means 4 or fewer: read the cumulative frequency of 4, which is 16.
The cumulative frequency of 5 is 23, so 23 households have 5 or fewer. More than 5: $30 - 23 = \mathbf{7}$ . (Check with the frequencies: $4 + 2 + 1 = 7$ . ✓)
Those with 6 or fewer, minus those with 2 or fewer: $27 - 3 = \mathbf{24}$ .
$\frac{16}{30} \times 100\% \approx 53.3\%$ .

Worked example 2: a missing frequency

Forty S1 learners took a quiz marked out of 10. One frequency was smudged:

Mark	4	5	6	7	8	9	10
Frequency	2	4	7	$k$	9	5	3

(a) Find $k$ . (b) Make the cumulative frequency row. (c) The pass mark is 6. How many learners passed?

Solution

$2 + 4 + 7 + k + 9 + 5 + 3 = 40$ , so $30 + k = 40$ and $k = 10$ .
Running totals: 2, 6, 13, 23, 32, 37, 40. The last entry is 40. ✓
A learner fails with a mark of 5 or less; the cumulative frequency of 5 is 6. So $40 - 6 = 34$ learners passed.

Worked example 3: from cumulative frequencies back to frequencies

A school football team recorded the goals it scored in each match of a season, but kept only the cumulative frequencies:

Goals	0	1	2	3	4
Cumulative frequency	4	11	19	26	30

Find the frequency of each number of goals.

Solution Undo the running total: each frequency is a cumulative frequency minus the one before it.

Goals	0	1	2	3	4
Frequency	4	$11 - 4 = 7$	$19 - 11 = 8$	$26 - 19 = 7$	$30 - 26 = 4$

Check: $4 + 7 + 8 + 7 + 4 = 30$ matches. ✓ The team played 30 matches.

Warning

Common mistakes

Counting a value twice or missing one. Tick each raw value off as you tally it, and always check $\sum f = n$ .
"More than" questions. The cumulative frequency counts values up to and including $x$ . For "more than $x$ ", subtract it from $n$ ; for "at least $x$ ", subtract the cumulative frequency of the value below $x$ .
Writing the cumulative frequency in place of the frequency. In Worked example 2, 23 learners scored 7 or less, but only 10 scored exactly 7.

Tip

Read a cumulative frequency aloud as a sentence: "16 households have 4 people or fewer." If the sentence sounds wrong, the number is being used wrongly.

FuseSchool: Frequency Tables (4 min). Watch how raw data are tallied into a frequency table, and how the total of the frequencies is checked.

✦

Practice

Questions 4 to 6 use the household table in this lesson.

The number of eggs laid each day by a farmer's hens over 20 days was: 5, 6, 4, 6, 7, 5, 6, 6, 8, 5, 7, 6, 4, 6, 5, 7, 6, 8, 5, 6. Make a tally and frequency table.
Add a cumulative frequency column to your table from question 1. On how many days were 6 or fewer eggs laid?
On how many days in question 1 were more than 6 eggs laid?
How many households have at least 5 people?
How many households have exactly 4 people? Which row of the table shows it?
What percentage of the households have 3 people or fewer? Give your answer to one decimal place.
The cumulative frequencies of the marks 5, 6, 7 and 8 in a test are 12, 20, 31 and 35, and nobody scored more than 8. Find the frequencies of the marks 6, 7 and 8.

Show answersHide answers

4: 2, 5: 5, 6: 8, 7: 3, 8: 2; total 20.
Cumulative frequencies 2, 7, 15, 18, 20; on 15 days.
$20 - 15 = 5$ days.
At least 5 means 5 or more: $30 - 16 = 14$ households.
8 households: the frequency column of the row $x = 4$ , not its cumulative frequency.
$\frac{8}{30} \times 100\% \approx 26.7\%$ .
6: $20 - 12 = 8$ ; 7: $31 - 20 = 11$ ; 8: $35 - 31 = 4$ .

Sources and credits

8.1: Gathering and Organizing Data (frequency distributions), from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Video: Frequency Tables, FuseSchool - Global Education, embedded from YouTube.

The cumulative frequency section, the Rwandan data, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1Thirty numbers in a notebook

2Building a frequency table

Thirty numbers in a notebook

Building a frequency table