Tally Charts and Frequency Tables

Use tally marks to turn a list into a frequency table, check the total, and fill in the gaps in a table with row and column totals.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 37 real past-paper questions below

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Tallies and Frequency

A tally chart helps you count a list without losing your place. Draw one line for each item. The fifth line goes across the first four, making a gate of 5.

You draw	It means
$∣∣∣$	3
$∣∣∣∣$	5
$∣∣∣∣ ∣∣$	7
$∣∣∣∣ ∣∣∣∣ ∣$	11

Rule

Frequency means how many times. Count the gates in fives, then add the single lines.

Key idea

In the 9th century in Iraq, the scholar al-Kindi broke secret codes with a frequency table. He counted how often each letter appears in ordinary Arabic, then counted the symbols in the secret message. The most common symbol probably stood for the most common letter.

From a List to a Frequency Table

The 2-mark question: “Complete the frequency table” from a list. 20 students said how they get to school:

bus, walk, walk, car, bike, walk, bus, walk, car, walk, bike, bus, walk, car, walk, bus, bike, walk, car, bus

Worked example

1
$bus \to bus row: ∣$ — Start at the first word. Draw one line in its row, then tick the word off the list so you never count it twice.
2
$walk: ∣∣∣∣ ∣∣∣$ — Carry on word by word to the end of the list. Every word gets exactly one line, in its own row.
3
$walk: 5 + 3 = 8$ — Now count each row. A gate is 5, and 3 single lines more makes 8. Write 8 in the Frequency column.
4
$bus: ∣∣∣∣ = 5$ — One full gate and no single lines, so 5.
5
$car: ∣∣∣∣ = 4$ — Four lines with nothing across them yet. The fifth line would have made a gate, so this is 4.
6
$bike: ∣∣∣ = 3$ — Three single lines, so 3.
7
$8 + 5 + 4 + 3 = 20$ — Check: the frequencies add up to the 20 students in the list, so nobody was missed or counted twice.

Common mistake

Leaving the Frequency column empty. Tallies alone lose a mark: always write each count as a number.

Key idea

The mode is the row with the biggest frequency: walk, with 8. The answer is walk, not 8.

Filling Gaps With Totals

The 3-mark question: a table with row and column totals, and some gaps. 50 students were asked. Every row and every column adds up to its total.

Travel	Boys	Girls
Walk	14	?
Bus	?	9
Total	25	?

Worked example

1
$Boys, bus = 25 - 14 = 11$ — The Boys column has only one gap. Its cells add up to the boys' total, 25, so take away the 14 boys who walk.
2
$Girls total = 50 - 25 = 25$ — The Girls column has two gaps, so it cannot be done yet. Use the grand total instead: all 50 students take away the 25 boys leaves the girls.
3
$Girls, walk = 25 - 9 = 16$ — Now the Girls column has only one gap. The girls' total take away the 9 girls on the bus gives the girls who walk.
4
$Walk: 14 + 16 = 30$ — Check the first row: boys who walk plus girls who walk gives everyone who walks.
5
$Bus: 11 + 9 = 20$ — Check the second row the same way.
6
$30 + 20 = 50$ — The rows add up to the grand total, so every gap is right.

Rule

Look for a row or column with only one gap. Fill it, then look again. If the facts come as sentences (“30 of the students are girls”), draw the table with a Total row and a Total column first, and put each number in its cell.

Common mistake

Adding across when the total is at the bottom. Check which way each total runs.

Missing Frequencies From the Total

Exams give a table with gaps and tell you the total. Everything you know adds up to the total, so the gaps share what is left.

Goals	0	1	2	3	4	5
Matches	6	11	9	6	?	?

There were 40 matches. Three times as many matches had 4 goals as had 5 goals.

Worked example

1
$6 + 11 + 9 + 6 = 32$ — Add the frequencies you know. These 32 matches are already counted.
2
$40 - 32 = 8$ — Take them away from the total. The 8 matches left are the 4-goal and 5-goal matches together.
3
$5 goals = n, 4 goals = 3 n$ — Call the smaller group n. Three times as many means the other group is 3n.
4
$n + 3 n = 4 n = 8$ — The two groups together make the 8 left over, and n + 3n is 4 equal parts.
5
$n = 8 \div 4 = 2$ — Share 8 into 4 equal parts. So 2 matches had 5 goals.
6
$3 \times 2 = 6$ — Three times as many had 4 goals, so 6 matches.
7
$32 + 6 + 2 = 40$ — Check: the whole table adds up to the total of 40.

Common mistake

Dividing 8 by 3 because the question says “three times”. The groups are n and 3n, which is 4 equal parts, so divide by 4.

Key idea

“Explain why the total cannot be 30.” The frequencies you know already add up to 32, which is more than 30. A frequency cannot be negative, so the total must be at least 32.

Key idea

Write the line 40 − 32 = 8 in your answer. It earns a mark for working even if the sharing goes wrong.

Fractions and Percentages of the Total

After the table, exams often ask “What fraction …?” or “What percentage …?”. Both use the frequency over the total.

Formula

fraction = \frac{frequency}{total}

Formula

percentage = \frac{frequency}{total} \times 100

frequency is the count for the group the question asks about. total is all the frequencies added together, or the total the question gives you.

A tally shows that 6 teachers and 18 students cycle to school. What fraction of the people who cycle are students? What percentage?

Worked example

1
$6 + 18 = 24$ — Find the total first: everyone who cycles, teachers and students together. It is not just the 18 students.
2
$\frac{18}{24}$ — Put the students on top, because the question asks about them. The total goes underneath.
3
$\frac{18 \div 6}{24 \div 6} = \frac{3}{4}$ — Divide top and bottom by 6, the biggest number that goes into both. This is the simplest form.
4
$\frac{3}{4} \times 100 = 75%$ — For the percentage, multiply the fraction by 100.

Common mistake

Writing $\frac{18}{6}$ , students over teachers. The bottom of the fraction must be the total, 24.

Key idea

Write $\frac{18}{24}$ before you simplify. It earns a mark for working even if the simplifying goes wrong.

Which Method? Read the Question

The table is the same idea every time: count, then check the total.

You see	You do
Complete the frequency table	tally the list in order, then write each count as a number
Tally column given	count gates in fives, add the lines
A frequency is missing	total take away the rest; share what is left
Row and column totals	one gap at a time, take away
What fraction or percentage	frequency over total; × 100 for a percentage
Most common	biggest frequency; give the item
Classes like 0 to 10	Grouped Frequency Tables
Draw a chart of it	Bar Charts and Composite Bar Charts
Find the mean from the table	Averages from Frequency Tables
Probability from a table	Probability from Tables and Charts

Key idea

The frequencies always add up to the number of items in the list. Check it before you move on.

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