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

Frequency Tables

Tally marks, class intervals and relative frequency — organising raw data into a clear picture in seconds.

Today’s hook: Your school tuck shop records every item sold. At the end of the day, the owner needs to know: how many meat pies? How many waters? A frequency table organises this chaos into a clear picture in seconds.

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Think First

warm-up

Here is a list of 15 scores: 7, 3, 5, 7, 4, 3, 7, 5, 6, 3, 4, 7, 5, 6, 3. Without organising them, how many 7s are there? Now try tallying them. Which method was faster and less error-prone?

Record your answer in your workbook.

The Big Idea

+5 XP

A frequency table counts how often each value or category occurs in a data set. It turns a messy list of raw data into an organised summary. The frequency of a value is the number of times it appears. All frequencies must add up to the total number of observations, n.

A basic frequency table has three columns: Category/Value, Tally, and Frequency. You go through the raw data once, adding a tally mark for each observation. Every fifth mark crosses the previous four (a “gate”), making it easy to count in fives. The frequency column counts the tally marks.

Sum of all frequencies = n (total data points)

List values first

Before tallying, write all possible values in the first column so none are missed.

Cross off as you go

Cross each data value off the raw list after tallying it to avoid double-counting.

Check the total

Always verify: sum of frequencies = n. If not, you missed or double-counted something.

What You'll Master

objectives

Know

What frequency and tally marks mean
The structure of a frequency table (value, tally, frequency)
What class intervals are and when to use them

Understand

Why frequencies must sum to n
How to choose equal class intervals for grouped data
How relative frequency converts frequency to a proportion or percentage

Can Do

Build a tally and frequency table from a raw data list
Create a grouped frequency table with equal class intervals
Calculate relative frequency as a decimal and as a percentage

Words You Need

vocabulary

FrequencyThe number of times a value or category appears in a data set.

Tally markA stroke used to count. Every fifth stroke crosses the previous four to make a “gate” of 5.

Class intervalA range of values grouped together in a frequency table (e.g. 10–19, 20–29).

Grouped dataData that has been sorted into class intervals rather than listed individually.

Relative frequencyFrequency ÷ total. Expresses each frequency as a proportion (decimal) or percentage.

Cumulative frequencyA running total of frequencies, adding each new class to the sum so far.

Spot the Trap

heads-up

Wrong: Writing tally marks as individual strokes without grouping: “IIIIIIII” for 8. You can’t quickly count this — the purpose of grouping is lost.

Right: Group in fives: IIII-I II I where the 5th stroke crosses the four. 8 = one gate (5) + three singles = IIII/ III. Count instantly: 5 + 3 = 8.

Wrong: Using unequal class intervals: 0–9, 10–14, 15–25. The intervals have different widths (10, 5, 11) making the table misleading and any bar chart drawn from it distorted.

Right: Use equal class intervals: 0–9, 10–19, 20–29, 30–39. Each interval covers exactly 10 values, making comparisons fair.

Making a Tally and Frequency Table

+5 XP

To make a frequency table from raw data: Step 1 list all possible values in column 1. Step 2 go through the data one item at a time, adding a tally mark. Step 3 count the tallies and write the frequency. Step 4 sum the frequencies and check they equal n.

Raw data (favourite colours, n = 12): Red, Blue, Green, Blue, Red, Red, Green, Blue, Red, Blue, Green, Blue. Go through each: Red appears 4 times, Blue 5 times, Green 3 times. Check: 4 + 5 + 3 = 12 = n. The table is complete. Order the categories before you start to avoid missing any.

List → Tally → Count → Check sum

One pass through the data

Go through your raw list exactly once. Cross off each value as you tally it.

Gates of 5

The crossing stroke on the 5th mark groups tallies so you can count in fives at a glance.

Sum check is mandatory

If your frequencies don’t sum to n, find the error before moving on.

Grouped Frequency Tables with Class Intervals

+5 XP

When data has many different values (e.g. scores from 0 to 100), listing every value individually creates a huge, unhelpful table. Instead, group the data into class intervals — equal-width ranges. A class interval width of 10 is common for scores; use wider intervals if there are fewer data points.

Scores: 45, 52, 61, 73, 48, 55, 67, 70, 42, 58, 63, 76. Class intervals of width 10: 40–49 (45, 48, 42 → 3), 50–59 (52, 55, 58 → 3), 60–69 (61, 67, 63 → 3), 70–79 (73, 70, 76 → 3). Total = 12 = n. Each interval is written with its lower and upper bounds.

Class interval width = (max − min) ÷ number of classes

5–10 classes is ideal

Too few classes hides the pattern; too many defeats the purpose of grouping.

No gaps, no overlaps

Intervals must cover every value with no gaps and no overlapping boundaries.

Equal widths only

All class intervals must have the same width. Mixed widths distort graphs.

Relative Frequency and Percentages

+5 XP

Relative frequency = frequency ÷ total (n). It expresses each category’s share as a proportion (decimal between 0 and 1) or as a percentage (multiply by 100). All relative frequencies must sum to 1 (or 100%). Relative frequency is useful when comparing groups of different sizes.

From the colour example (n = 12): Red = 4, Blue = 5, Green = 3. Relative frequencies: Red: 4 ÷ 12 = 0.333 = 33.3%, Blue: 5 ÷ 12 = 0.417 = 41.7%, Green: 3 ÷ 12 = 0.250 = 25.0%. Check: 0.333 + 0.417 + 0.250 = 1.000. Due to rounding, allow ±0.01.

Relative frequency = f ÷ n | Percentage = (f ÷ n) × 100

Sum must be 1

All relative frequencies must sum to 1.00 (allowing tiny rounding differences).

Compare across groups

Relative frequency lets you compare classes of different sizes fairly.

3 decimal places

Round relative frequencies to 3 decimal places (or 1 decimal place for %).

Watch Me Solve It · 3 examples

Watch Me Solve It · Build a frequency table

+15 XP per step

PROBLEM

Build a frequency table for this data (number of pets): 2, 0, 1, 3, 2, 1, 0, 2, 1, 2, 3, 1, 0, 2, 1.

1

List all possible values

Minimum = 0, maximum = 3. Values: 0, 1, 2, 3.

List all values from smallest to largest before you start tallying.
2

Tally each value

0: III (3 times) 1: IIII I (5 times) 2: IIII I (5 times) 3: II (2 times)
3

Write frequencies and check sum

0→3, 1→5, 2→5, 3→2. Sum = 3 + 5 + 5 + 2 = 15 = n. Correct!

The sum equals 15 (the number of data values), so the table is complete.

Answer0: 3, 1: 5, 2: 5, 3: 2 (Total = 15)

Watch Me Solve It · Grouped frequency table

+15 XP per step

PROBLEM

Group these 12 test scores into class intervals of width 10: 34, 56, 72, 45, 61, 38, 55, 79, 43, 68, 52, 77.

1

Find range and set up intervals

Min = 34, Max = 79. Intervals: 30–39, 40–49, 50–59, 60–69, 70–79.

Start the first interval below the minimum; end the last above the maximum.
2

Sort each score into its interval

30–39: 34, 38 → 2 40–49: 45, 43 → 2 50–59: 56, 55, 52 → 3 60–69: 61, 68 → 2 70–79: 72, 79, 77 → 3
3

Write the table and check

Frequencies: 2, 2, 3, 2, 3. Sum = 2 + 2 + 3 + 2 + 3 = 12 = n. Correct!

Each score belongs to exactly one interval. No score is left out.

Answer30–39: 2, 40–49: 2, 50–59: 3, 60–69: 2, 70–79: 3 (Total = 12)

Watch Me Solve It · Relative frequency

+15 XP per step

PROBLEM

A class of 30 students was asked their favourite subject. Results: Maths 12, English 8, Science 10. Calculate relative frequency (decimal) and percentage for each.

1

Check the total

12 + 8 + 10 = 30 = n. Total confirmed.
2

Calculate relative frequency (f ÷ n)

Maths: 12 ÷ 30 = 0.400 English: 8 ÷ 30 = 0.267 Science: 10 ÷ 30 = 0.333
3

Convert to percentages and check

Maths: 40.0% English: 26.7% Science: 33.3%. Sum = 100% ✓

Rounding to 1 decimal place: 40.0 + 26.7 + 33.3 = 100.0%. Always check.

AnswerMaths: 0.400 (40%) English: 0.267 (26.7%) Science: 0.333 (33.3%)

Common Pitfalls

heads-up

Forgetting the fifth tally mark crosses the group

Some students write 5 separate strokes (IIIII) instead of a crossed gate. The gate pattern (four strokes then a diagonal cross) is specifically designed so you can count in fives instantly. Without it you have to count each stroke individually.

Fix: On every 5th data point, draw a diagonal line crossing the four strokes. Practice this until it is automatic.

Using unequal class intervals

If one interval covers 5 values and another covers 20, a bar chart drawn from the table will be misleading — the wide interval will appear to have fewer data points per unit of range than it really does.

Fix: Choose a width and apply it consistently: 0–9, 10–19, 20–29 … Never vary the width mid-table.

Forgetting that frequencies must sum to n

This check reveals every error: if you double-counted a value, the sum exceeds n; if you missed a value, the sum is less than n. Many students skip this check and submit tables with errors.

Fix: Always write “Total = ” at the bottom of the frequency column and verify it equals n before moving on.

Copy Into Your Books

Frequency Table Steps

List all values (or intervals) in column 1
Tally each data point (cross off as you go)
Count tallies → write frequency
Check: Σf = n

Tally Mark Rules

4 strokes then a diagonal cross = 5
Count in fives + remainder
8 = IIII/ III (gate of 5 + 3)

Class Intervals

Equal width only (e.g. all width 10)
5–10 classes is ideal
No gaps, no overlaps

Relative Frequency

Relative freq = f ÷ n (decimal)
Percentage = (f ÷ n) × 100
All relative freqs sum to 1

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Frequency Tables

4 problems

Four drill problems. Work each out before revealing the answer.

1 Data: 3, 5, 3, 7, 5, 3, 5, 7, 3, 5, 7, 5, 3, 5, 7. Build a tally and frequency table.

3: IIII I → 5 | 5: IIII II → 6 | 7: IIII → 4. Total: 5 + 6 + 4 = 15 = n. (Mode = 5 with frequency 6.)3→5, 5→6, 7→4, total = 15
2 From the table above, what is the frequency of scores 10–14 if the data were: 10–14: 6 out of 30?

Frequency of 10–14 = 6. The 6 students whose scores fell between 10 and 14 inclusive are counted in this class interval.Frequency = 6
3 6 out of 30 students chose pizza. What is the relative frequency?

Relative frequency = 6 ÷ 30 = 0.2. As a percentage: 0.2 × 100 = 20%.0.2 or 20%
4 You are grouping ages from 11 to 17. What class interval width would you choose and what classes would you use?

Width 2 works well (4 classes): 11–12, 13–14, 15–16, 17–18. Or width 3 (3 classes): 11–13, 14–16, 17–19. Both are equally valid; choose based on sample size. Avoid width 1 (too many classes for only 7 possible ages).e.g. 11–12, 13–14, 15–16, 17–18 (width = 2)

Complete in your workbook.

Quick Check · 5 questions

The tally IIII/ III represents what frequency?

+10 XP

A frequency table has frequencies 6, 9, 5, 10. What is n?

+10 XP

8 students out of 40 chose basketball. What is the relative frequency?

+10 XP

Which is most important when setting up class intervals for grouped data?

+10 XP

A score of 25 would be placed in which class interval?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Build a tally and frequency table for this raw data (type of transport to school): bus, walk, bus, car, bike, walk, bus, walk, car, bus, walk, bus, car, walk, bus. Show all tally marks and verify your total.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. A frequency table shows: 10–14 (freq 4), 15–19 (freq 7), 20–24 (freq 9), 25–29 (freq 5). (a) What is n? (b) What is the relative frequency of the 15–19 class?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. A student groups the ages of 20 survey respondents into intervals: 10–14 (5 people), 15–24 (8 people), 25–39 (7 people). (a) What is wrong with these intervals? (b) What is a better set of intervals? (c) Recalculate the relative frequencies using equal intervals of width 10 (assume the same data fits into 10–19, 20–29, 30–39 as 9, 6, 5).

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — 8. One gate (5) + three singles (3) = 8.

2. B — 30. n = 6 + 9 + 5 + 10 = 30.

3. A — 0.2 (20%). Relative frequency = 8 ÷ 40 = 0.2.

4. D — Equal class interval widths. Ensures fair comparison and undistorted graphs.

5. C — 20–29. 25 falls between 20 and 29 inclusive.

Show Your Working Model Answers

Q6 (3 marks): Bus: IIII/ I → 6 [1]; Walk: IIII/ → 5 [accept 5]; Car: III → 3; Bike: I → 1. Total = 6 + 5 + 3 + 1 = 15 = n [1 for correct tallies, 1 for correct total].

Q7 (2 marks): (a) n = 4 + 7 + 9 + 5 = 25 [1]. (b) Relative frequency of 15–19 = 7 ÷ 25 = 0.28 (28%) [1].

Q8 (4 marks): (a) Unequal class widths (5, 10, 15) — makes comparison unfair and graphs misleading [1]. (b) Equal-width intervals such as 10–19, 20–29, 30–39 (width = 10) [1]. (c) n = 9 + 6 + 5 = 20. 10–19: 9 ÷ 20 = 0.45 (45%) [1]; 20–29: 6 ÷ 20 = 0.30 (30%); 30–39: 5 ÷ 20 = 0.25 (25%). Check: 0.45 + 0.30 + 0.25 = 1.00 [1].

Stretch Challenge · +25 XP, +10 coins

The Data Organiser

A class of 25 students recorded their daily screen time (hours) rounded to the nearest hour: 3, 5, 2, 4, 6, 3, 5, 7, 2, 4, 5, 3, 6, 4, 5, 2, 3, 5, 4, 6, 3, 5, 2, 4, 7. (a) Build a complete frequency table. (b) Add a relative frequency column (to 3 decimal places). (c) What percentage of students had more than 4 hours of screen time? (d) If the recommended maximum is 4 hours, what proportion exceeded this?

Reveal solution

Tally: 2→4, 3→5, 4→5, 5→6, 6→3, 7→2. Total = 25. (b) Relative frequencies: 2: 0.160, 3: 0.200, 4: 0.200, 5: 0.240, 6: 0.120, 7: 0.080. Sum = 1.000. (c) Above 4 hours (5, 6, or 7): 6 + 3 + 2 = 11 students. 11 ÷ 25 = 0.44 = 44%. (d) Proportion = 11/25 = 0.44.

Quick Review

Frequency table steps

List → Tally → Count → Check Σf = n

Tally marks

Groups of 5: four strokes then a diagonal cross

Class intervals

Equal width, no gaps, no overlaps, 5–10 classes

Relative frequency

f ÷ n (decimal) or (f ÷ n) × 100 (percentage)

Sum check

Relative frequencies must sum to 1.000 (allow tiny rounding)

When to group

Many different values → group. Few distinct values → list each.

Interactive: Frequency Table Builder

Enter a list of raw data values and watch a live frequency table and bar chart build themselves. Adjust the class interval width and see how the table changes.

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