Lesson 10 ~25 min Unit 4 · Data & Chance +85 XP

The Mode and Range

Find the most frequent value and measure total spread — two simple but essential statistical tools for any dataset.

Today's hook: The most popular pizza flavour at your school canteen isn’t an “average” — it’s the MODE. Meanwhile the range tells you whether test scores were packed tightly together or wildly spread out. Two simple but essential statistical tools.

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Think First

warm-up

Before you read on — quickly: A shoe shop sells sizes 7, 8, 7, 9, 7, 8, 10, 7, 8. The manager needs to order more stock. Should she order the mean shoe size, the median shoe size, or the most popular shoe size? Why? Try it, then check your reasoning as you go.

Record your answer in your workbook.

The Big Idea

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The mode is the value that occurs MOST often in a dataset. It can have none, one, or multiple modes. The range measures total spread: Range = Maximum − Minimum. Together they answer “what is most popular?” and “how spread out is the data?”

Dataset: 5, 8, 3, 8, 7, 5, 8, 3, 5. Count each value: 3→2, 5→3, 7→1, 8→3. Both 5 and 8 appear 3 times — this is bimodal. Range = 8 − 3 = 5. The range tells you the data spans 5 units. Mode tells you popularity; range tells you spread.

Mode = most frequent Range = Max − Min

Count every value

Tally each unique value before deciding which is most frequent.

Mode is the VALUE, not frequency

If 5 appears 3 times, the mode is 5 — not 3. Never confuse these.

Range = subtract, not add

Range = Max − Min. Always subtract the smallest from the largest.

What You'll Master

objectives

Know

Mode = value that appears most often
Range = Maximum − Minimum
A dataset can have no mode, one mode, or many modes

Understand

Why bimodal and multimodal datasets arise
What a large or small range tells you about spread
Why the mode is appropriate for categorical data

Can Do

Find the mode from a list or frequency table
Calculate range and interpret it in context
Choose between mean, median and mode for any scenario

Words You Need

vocabulary

ModeThe value that appears most often in a dataset.

Modal classThe category or class interval with the highest frequency.

BimodalA dataset with exactly two modes (two values tied for most frequent).

MultimodalA dataset with more than two modes.

No modeWhen every value appears the same number of times — no single most frequent value.

RangeRange = Maximum − Minimum. Measures the total spread of data.

Spot the Trap

heads-up

Wrong: Listing the FREQUENCY as the mode. For {3,3,5,5,5}, the mode is 5 (appears 3 times) — NOT 3 (its frequency).

Right: The mode is always the DATA VALUE with the highest count, not the count itself. Say “mode = 5” and note “it appears 3 times” separately.

Wrong: Saying “no mode” when values appear the same number of times but SOME appear more than others. Only say “no mode” when all values appear equally.

Right: Check ALL frequencies. If any value has a higher count than the rest, that is the mode. “No mode” only applies when every value appears the same number of times.

Finding Mode from a List and Frequency Table

+5 XP

From a raw list: tally how often each value appears — the highest tally wins. From a frequency table: scan the frequency column for the largest number, then read the corresponding value.

List: 5, 8, 3, 8, 7, 5, 8, 3, 5. Tally: 3→2, 5→3, 7→1, 8→3. Highest count = 3, shared by 5 and 8. This dataset is bimodal — modes are 5 and 8. From a frequency table: find the row with the largest frequency, then read its value (not the frequency number).

Mode = VALUE with highest frequency (not the count itself)

Tally all values

Go through the list systematically. Skipping values leads to the wrong mode.

Report all modes

If two or more values tie for highest frequency, report ALL of them as modes.

Frequency table shortcut

In a frequency table, scan the f-column for the max, then look left for the value.

Calculating Range and Interpreting Spread

+5 XP

The range is a measure of spread, not centre. Range = Maximum − Minimum. A large range means data is widely spread; a small range means values cluster together. The range uses only the two extreme values, so one outlier can inflate it dramatically.

Scores: 23, 45, 12, 67, 34. Maximum = 67. Minimum = 12. Range = 67 − 12 = 55. In context: “The scores spread across a range of 55 marks.” Compare two classes with the same mean: Class A range = 10 (consistent), Class B range = 80 (wildly variable) — the range reveals what the mean hides.

Range = 67 − 12 = 55

Find max and min first

Scan the data for the highest and lowest values before subtracting.

Range is spread, not centre

Range cannot tell you the typical value — only how far apart the extremes are.

Outliers inflate range

One extreme value makes the range large. It is not a robust measure of spread.

Choosing Between Mean, Median and Mode

+5 XP

Each measure of centre has its best context. Categorical data (colours, flavours, sizes) → only mode works. Skewed data or outliers → median. Symmetric data, no outliers → mean.

Scenario: favourite pizza toppings in a class. Toppings are categories — you cannot average “Hawaiian” and “Margherita”. Only the mode (most popular topping) makes sense. Now if data is exam scores with one very high score (outlier), use the median. If scores are symmetric with no outliers, use the mean.

Mode for categories · Median for skew · Mean for balance

Categories → mode

Shoe sizes, pizza flavours, colours — you can’t average words, only count them.

Outliers → median

House prices, incomes — a few extreme values make the mean misleading.

Balanced → mean

Heights, daily temperatures — symmetric data with no extreme outliers.

Watch Me Solve It · 3 examples

Watch Me Solve It · Finding a bimodal mode

+15 XP per step

PROBLEM

Find the mode(s) of: 5, 8, 3, 8, 7, 5, 8, 3, 5.

1

Tally each value’s frequency

3 → 2 times 5 → 3 times 7 → 1 time 8 → 3 times

Go through the list carefully. Every value must be counted.
2

Find the highest frequency

Highest frequency = 3. Both 5 and 8 appear 3 times.
3

State the modes

This dataset is bimodal. Modes = 5 and 8.

Report all modes. Never report just one when two values tie. The mode is the VALUE (5 and 8), not the count (3).

AnswerBimodal: modes are 5 and 8

Watch Me Solve It · Range with context

+15 XP per step

PROBLEM

Calculate the range of: 23, 45, 12, 67, 34. Interpret the result in context.

1

Identify the maximum and minimum

Maximum = 67 Minimum = 12

Scan all values. The range only uses these two extreme values.
2

Calculate the range

Range = 67 − 12 = 55
3

Interpret in context

The scores spread across 55 marks — quite a wide variation in performance.

A range of 55 on a typical 100-mark test suggests the group is not performing consistently.

AnswerRange = 55. Scores spread across 55 marks.

Watch Me Solve It · Choosing the right measure

+15 XP per step

PROBLEM

A survey records favourite pizza toppings in a class. Which measure of centre is appropriate? Then separately: exam scores include one very high outlier. Which measure then?

1

Pizza toppings: identify the data type

Toppings (Hawaiian, Margherita, BBQ…) are categorical. You cannot average category names.

Mean and median require numerical ordering. Categories only support counting — so mode is the only valid measure.
2

Pizza toppings: answer

Use the mode — it identifies the most popular (most frequently chosen) topping.
3

Exam scores with outlier: answer

Use the median — the outlier inflates the mean, making it unrepresentative. Median is resistant.

One very high score raises the mean well above most students’ actual performance. Median ignores the magnitude of that outlier.

AnswerPizza toppings → Mode. Exam scores with outlier → Median.

Common Pitfalls

heads-up

Confusing mode value with its frequency

For {3,3,5,5,5}: the mode is 5 (appears 3 times). Students often write “mode = 3” confusing the frequency (3 times) with a value. Or they write “mode = 3” because they see 3 appear first. Always identify the VALUE with the highest count.

Fix: After tallying, ask “which VALUE appears most?” not “what is the highest count?”.

Saying “no mode” when values appear equally vs. all different

{2, 4, 4, 6, 6, 8}: 4 and 6 both appear twice. This is bimodal — NOT “no mode”. “No mode” only applies when every single value has the same frequency (e.g. each appears once).

Fix: “No mode” only if all frequencies are identical. Any standout frequency creates a mode.

Range is affected by outliers — it is not robust

Scores: 50, 52, 54, 56, 100. Range = 100−50 = 50. Without the outlier: range = 56−50 = 6. One extreme value makes the range jump from 6 to 50. Never interpret range as “typical spread” when outliers may exist.

Fix: Mention outliers when reporting range. “Range = 50, but this is largely due to the outlier value of 100.”

Copy Into Your Books

The Mode

Mode = value with the highest frequency
Can have no mode, one mode, or many
Bimodal = 2 modes tied for highest
Report the VALUE, not the count

The Range

Range = Maximum − Minimum
Measures spread, not centre
Large range = widely spread data
One outlier can inflate range greatly

Choose the Right Measure

Mean: symmetric, no outliers
Median: skewed or outliers present
Mode: categorical data or “most popular”

Spot the Traps

Mode = value, not frequency count
“No mode” only when all frequencies equal
Range affected by outliers
Categories: mode only (no mean/median)

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mode and Range

4 problems

Four drill problems to sharpen your mode and range skills. Work each, then reveal the answer.

1 Mode of: 5, 8, 3, 8, 7, 5, 8, 3, 5

Tally: 3→2, 5→3, 7→1, 8→3. Both 5 and 8 appear 3 times.Bimodal: modes are 5 and 8
2 Mode of shoe sizes: 7, 8, 9, 7, 8, 7, 9, 8, 7

Tally: 7→4, 8→3, 9→2. Highest count is 4 for size 7.Mode = 7
3 Range of: 23, 45, 12, 67, 34

Max = 67, Min = 12. Range = 67 − 12.Range = 55
4 Would you use mean or mode to describe the most popular YouTube category? Why?

YouTube categories are categorical (Gaming, Music, etc.). You cannot calculate a mean of labels. The mode identifies which category appears most — i.e. the most popular.Mode (categories cannot be averaged)

Complete in your workbook.

Quick Check · 5 questions

What is the mode of: 4, 7, 2, 7, 9, 4, 7?

+10 XP

What is the range of: 15, 8, 23, 4, 19?

+10 XP

A dataset is bimodal. How many modes does it have?

+10 XP

Which measure of centre is appropriate for favourite colours?

+10 XP

Dataset: 5, 6, 7, 8, 9. What is the range?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. A frequency table shows: Score 2 (freq 4), Score 3 (freq 7), Score 4 (freq 5), Score 5 (freq 3), Score 6 (freq 7). Find the mode(s) and the range of scores.

Answer in your workbook.

Understand Easy 2 MARKS

Q7. Explain what it means when a dataset has “no mode”. Give an example of a dataset with no mode and one with two modes.

Answer in your workbook.

Reason Hard 4 MARKS

Q8. Team A scores over 5 games: 18, 20, 19, 21, 22. Team B scores: 5, 30, 15, 28, 22. Both teams have the same mean of 20. Compare their mode(s) and ranges. Which team is more consistent? Use the range (not mean) to justify your answer.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — 7. Tally: 7 appears 3 times, more than any other value.

2. B — 19. Max = 23, Min = 4. Range = 23−4 = 19.

3. B — 2. Bimodal means exactly two modes.

4. A — Mode. Colours are categorical; only the mode applies.

5. C — 4. Range = 9−5 = 4.

Show Your Working Model Answers

Q6 (3 marks): Highest frequency = 7, shared by Score 3 and Score 6 [1]. Bimodal: modes = 3 and 6 [1]. Range = 6−2 = 4 [1].

Q7 (2 marks): No mode means every value appears the same number of times — no single value occurs more than any other [1]. No mode example: {2, 5, 7, 11} (all appear once). Two modes example: {3, 3, 5, 5, 7} (3 and 5 both appear twice) [1].

Q8 (4 marks): Team A: no mode (all different), range = 22−18 = 4 [1]. Team B: no mode (all different), range = 30−5 = 25 [1]. Team A is more consistent [1]. Although both teams average 20, Team A’s scores span only 4 points, staying close to the mean each game. Team B’s scores span 25 points — performance varies wildly. The range reveals this inconsistency that the mean hides entirely [1].

Stretch Challenge · +25 XP, +10 coins

The Dataset Inventor

Create a dataset of 10 values where: the mode is 8, the range is 12, and the mean is NOT 8. Show that all three conditions are satisfied with full calculations.

Reveal solution

Example: {3, 5, 8, 8, 8, 8, 9, 10, 12, 15}. Mode = 8 (appears 4 times — most frequent) ✓. Max = 15, Min = 3. Range = 15−3 = 12 ✓. Mean = (3+5+8+8+8+8+9+10+12+15)÷10 = 86÷10 = 8.6 ≠ 8 ✓. Many valid datasets exist — you need at least two 8s (more than any other value), a difference of 12 between max and min, and a mean that is not exactly 8.

Quick Review

Mode = most frequent

Count each value and identify the one (or ones) with the highest tally.

Can have no/one/many modes

No mode: all equal frequency. Bimodal: two tie. Multimodal: three or more.

Range = Max − Min

Subtract the smallest from the largest. Always use subtraction, not addition.

Use mode for categories

Colours, flavours, sizes — you can only count categories, not average them.

Small range = consistent

A team with range 3 is far more predictable than one with range 40.

Choose the right measure

Mode for categories · Median for skew · Mean for balanced numerical data.

Interactive: Mode and Range Explorer

Enter your own data and see how the mode and range change — especially when you add repeated values or extreme outliers.

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