Lesson 3 ~25 min Unit 4 · Data & Probability +85 XP

Frequency Tables

Turn raw data into organised tables using tally marks. Add relative and cumulative frequency columns to unlock powerful summaries of any data set.

Today’s hook: Imagine 25 students each shout out their favourite colour at the same time. Chaos! A frequency table organises that chaos into a single clean summary in under a minute. Tallying is one of the oldest data tools in human history — archaeologists have found tally marks on bones that are 30,000 years old.

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Think First

warm-up

If 25 students each wrote down their favourite colour, how would you organise all those answers quickly so you could see which colour was most popular?

Record in your workbook.

The Big Idea

+5 XP

A frequency table organises raw data into categories with tally marks and counts. It is the foundation for every graph we draw in statistics.

Each row of the table represents one category or value. The tally records each data value as a mark. After tallying, count the marks to get the frequency — the number of times that value occurs.

Category | Tally | Frequency | Total

What You’ll Master

objectives

Know

How to construct a frequency table using tally marks
The formula for relative frequency
How to build a grouped frequency table with class intervals
How to add a cumulative frequency column

Understand

Why relative frequency is more useful than frequency when comparing groups of different sizes
Why class intervals for continuous data must be equal-width and non-overlapping

Can Do

Tally raw data and produce a complete frequency table
Calculate relative frequency as a fraction, decimal, and percentage
Identify the modal class from a grouped frequency table
Use cumulative frequency to answer “how many scored below X?”

Words You Need

vocabulary

Frequency ($f$)The number of times a value or category occurs in the data set.

TallyMarks used to record each data value as it occurs. Groups of 4 with a diagonal cross for the 5th: IIII̅.

Relative frequencyFrequency expressed as a proportion of the total: $f_{rel} = f \div n$. Can be a fraction, decimal, or percentage.

Cumulative frequencyThe running total of frequencies from the top of the table to the current row.

Class intervalA range of values used to group continuous data (e.g. 10–<20). Must be equal width and non-overlapping.

Modal classThe class interval with the highest frequency in a grouped frequency table.

Spot the Trap

heads-up

Wrong tally grouping: Writing 6 as “|||| ||” correctly, but tallying to 7 and calling it “|||| |||” is eight marks not seven — count carefully!

Right: Group the first 5 as |||| with a diagonal cross, then start a new group. Count groups-of-5 first, then add remaining marks.

Wrong class intervals: “0–10 / 10–20 / 20–30” — the value 10 could go in the first or second interval.

Right: Use “0–<10 / 10–<20 / 20–<30” so every value falls in exactly one interval (10 goes in the second).

Building a Frequency Table

+5 XP

Steps to build a frequency table:

List all possible categories or values in the first column.
Go through the raw data one value at a time, adding a tally mark to the correct row.
After tallying all data, count the marks to get the frequency for each row.
Add a Total row at the bottom. Check: the frequencies must sum to $n$ (total count).

Tally rule

First four marks are vertical (||||). The fifth is a diagonal cross. This makes groups of 5 easy to count.

Mode from table

The category with the highest frequency is the mode (or modal class for grouped data).

Relative Frequency

+5 XP

Relative frequency expresses each frequency as a proportion of the total. This allows fair comparison between data sets of different sizes.

$$f_{rel} = \frac{f}{n}$$

where $f$ is the frequency of one category and $n$ is the total number of data values.

As a fraction

Soccer in our example: $\dfrac{8}{20} = \dfrac{2}{5}$

As a decimal

$8 \div 20 = 0.4$

As a percentage

$0.4 \times 100 = 40\%$

Check: All relative frequencies must sum to exactly 1 (or 100%). If they don’t, recheck your arithmetic.

Grouped Frequency Tables

+5 XP

When data is continuous or has many values, use class intervals of equal width. Each interval covers a range of values.

Example: heights of 20 students grouped in 10 cm intervals.

Height (cm)	Tally	Freq	Class centre
150–<160	\|\|\|\|	4	155
160–<170	\|\|\|\| \|\|\|	8	165
170–<180	\|\|\|\| \|	6	175
180–<190	\|\|	2	185
Total		20

The class centre is the midpoint of each interval: $\dfrac{\text{lower} + \text{upper}}{2}$. Used when estimating the mean. Modal class: 160–<170 (highest frequency = 8).

Cumulative Frequency

+5 XP

Cumulative frequency is the running total — add each new frequency to all the previous ones. It answers questions like “How many scored less than 70?”

Score	Freq ($f$)	Cumulative freq
0–<20	3	3
20–<40	5	8 (3+5)
40–<60	9	17 (8+9)
60–<80	7	24 (17+7)
80–100	4	28 (24+4)

From this table: 17 students scored less than 60. The last cumulative frequency (28) equals the total $n = 28$. This is your check.

Watch Me Solve It · Estimated mean from grouped table

Watch Me Solve It · Relative frequency

+15 XP per step

PROBLEM

A class of 20 students chose a favourite sport. Results: Soccer 8, Basketball 6, Swimming 4, Other 2. Calculate the relative frequency for each sport as a fraction, decimal, and percentage.

1

Confirm total $n$

$n = 8 + 6 + 4 + 2 = 20$

Always verify the total before calculating relative frequencies.
2

Apply the formula $f_{rel} = f \div n$ to each category

Soccer: $8 \div 20 = 0.4 = 40\%$ Basketball: $6 \div 20 = 0.3 = 30\%$ Swimming: $4 \div 20 = 0.2 = 20\%$ Other: $2 \div 20 = 0.1 = 10\%$
3

Check: relative frequencies sum to 1

$0.4 + 0.3 + 0.2 + 0.1 = 1.0$ ✓

Or as percentages: $40 + 30 + 20 + 10 = 100\%$ ✓

AnswerSoccer 40%, Basketball 30%, Swimming 20%, Other 10%

Brain Trainer · 4 problems

Brain Trainer · Frequency Tables

4 problems

1 In a frequency table with $n = 40$, category A has frequency 10. What is its relative frequency as a decimal and percentage?

$f_{rel} = 10 \div 40 = 0.25 = 25\%$0.25 or 25%
2 A grouped table has class intervals 0–<10, 10–<20, 20–<30. What is the class centre of each interval?

Class centre = (lower + upper) ÷ 2. First: (0+10) ÷ 2 = 5. Second: (10+20) ÷ 2 = 15. Third: (20+30) ÷ 2 = 25.5, 15, 25
3 From a cumulative frequency table: 0–<10 has cumfreq 4, 10–<20 has cumfreq 11, 20–<30 has cumfreq 18. How many values are in the 10–<20 class?

Frequency = cumfreq at row − cumfreq at previous row = 11 − 4 = 7.7 values
4 A table has classes: 10–<20 (f=3), 20–<30 (f=8), 30–<40 (f=5). What is the modal class?

The modal class is the one with the highest frequency. 20–<30 has f = 8 which is the largest.Modal class: 20–<30

Common Pitfalls

heads-up

Overlapping class intervals

Writing “0–10, 10–20” means the value 10 belongs to two classes. This causes a counting error.

Fix: Write “0–<10, 10–<20” (using “less than” to make intervals non-overlapping).

Forgetting to check the total

If frequencies don’t sum to $n$, or relative frequencies don’t sum to 1, there is a tallying error.

Fix: Always include a Total row and verify it matches the number of data values.

Copy This Into Your Book

Frequency Table

List categories → tally each value → count to get frequency
Total row: sum of all frequencies = $n$
Mode = category with highest frequency

Relative Frequency

$f_{rel} = f \div n$
Express as fraction, decimal, or %
Check: all $f_{rel}$ sum to 1 (or 100%)

Grouped Table

Equal-width, non-overlapping intervals (use <)
Class centre = (lower + upper) ÷ 2
Modal class = interval with highest frequency

Cumulative Frequency

Running total: add each $f$ to all previous
Last row = $n$ (your check)
Used to find “how many scored below X?”

Quick Check · 5 questions

A frequency table shows: Soccer 8, Basketball 6, Swimming 4, Other 2, Total 20. How many students chose Soccer?

+10 XP

Using the same table (total = 20), what is the relative frequency of Basketball (frequency = 6)?

+10 XP

Which set of class intervals is correctly written to avoid overlap?

+10 XP

A grouped table has frequencies 3, 5, 9, 7, 4. What is the cumulative frequency after the 3rd row?

+10 XP

In a frequency table with 4 categories, the relative frequencies are 0.4, 0.3, 0.2, and 0.05. What does this tell us?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. The following 20 values were recorded (scores on a 1–5 scale): 3, 1, 4, 2, 5, 3, 3, 2, 4, 1, 5, 3, 2, 4, 3, 1, 2, 5, 4, 3. Construct a frequency table with columns for Score, Tally, Frequency, and Relative Frequency (as a decimal).

Draw the table in your workbook.

Apply Medium 3 MARKS

Q7. The heights (cm) of 15 students are: 152, 168, 175, 161, 158, 172, 164, 155, 179, 163, 170, 157, 165, 148, 169. Construct a grouped frequency table using intervals 145–<155, 155–<165, 165–<175, 175–<185. State the modal class.

Draw the table in your workbook.

Interpret Medium 3 MARKS

Q8. A cumulative frequency table for test scores shows: 0–<20 cumfreq 5, 20–<40 cumfreq 11, 40–<60 cumfreq 22, 60–<80 cumfreq 28, 80–100 cumfreq 30. (a) How many students scored less than 60? (b) How many scored in the 40–<60 class? (c) What is $n$?

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — Soccer frequency = 8.

2. C — $6 \div 20 = 0.3 = 30\%$.

3. A — “0–<10, 10–<20, 20–<30” is non-overlapping.

4. D — Cumfreq after row 3 = $3 + 5 + 9 = 17$.

5. B — Sum = 0.95 ≠ 1, so there is an error.

Model Answers

Q6 (3 marks): Score 1: f=3, rel=0.15; Score 2: f=4, rel=0.20; Score 3: f=6, rel=0.30; Score 4: f=4, rel=0.20; Score 5: f=3, rel=0.15; Total: 20, 1.00. [1 for correct frequencies; 1 for correct relative frequencies; 1 for total check].

Q7 (3 marks): 145–<155: f=2 (148, 152); 155–<165: f=6 (155, 157, 158, 161, 163, 164); 165–<175: f=5 (165, 168, 169, 170, 172); 175–<185: f=2 (175, 179). [1 for correct intervals; 1 for correct tallying; 1 for modal class]. Modal class: 155–<165 (f = 6).

Q8 (3 marks): (a) 22 students scored less than 60 (read cumfreq at end of 40–<60 row) [1]. (b) Frequency in 40–<60 = cumfreq 22 − cumfreq 11 = 11 students [1]. (c) $n$ = last cumulative frequency = 30 [1].

Stretch Challenge · +25 XP, +10 coins

Estimate the Mean from a Grouped Table

A class of 28 students sat a test. The grouped frequency table shows: 0–<40: $f=4$, 40–<60: $f=8$, 60–<80: $f=12$, 80–100: $f=4$. Use class centres to estimate the mean score. State the modal class. Show all working.

Reveal solution

Class centres: 0–<40 → 20; 40–<60 → 50; 60–<80 → 70; 80–100 → 90.
Sum of (centre × freq): $(20 \times 4) + (50 \times 8) + (70 \times 12) + (90 \times 4) = 80 + 400 + 840 + 360 = 1680$.
Estimated mean $= 1680 \div 28 = 60$.
Modal class: 60–<80 ($f = 12$, the highest).

Quick Review

Tally rule

4 vertical, 5th diagonal cross. Groups of 5.

Relative frequency

$f \div n$ — express as fraction, decimal, or %

Check sum

All relative frequencies must sum to 1 (100%)

Class intervals

Equal width, non-overlapping (use < notation)

Modal class

The class interval with the highest frequency

Cumulative frequency

Running total; last value = n

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Tally Tracker

Frequency Finder

Table Builder

Relative Reckoner

Cumulative Calculator

Data Organiser

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