How to Solve: Mean, Median, Mode, and Range from a List or Table

How to Solve: Mean, Median, Mode, and Range from a List or Table

GCSE & A-Level Maths

Introduction

"Mastering mean, median, mode, and range doesn’t just get you easy marks—it’s the key to unlocking real-world data questions worth 10-15% of your GCSE or A-Level Maths exam. One slip-up here could cost you a grade. Let’s make sure that doesn’t happen."

What You Need To Know First

Before diving in, you must already understand: 1. Ordering numbers – How to arrange data from smallest to largest. 2. Basic arithmetic – Addition, division, and counting. 3. Frequency tables – How to read and interpret them (for A-Level).

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Organise the Data

• If given a list, write the numbers in order (smallest to largest).
• If given a table, extract the numbers (ignore frequencies for now).

Step 2: Find the Range

• Identify the largest and smallest numbers.
• Subtract: Range = Largest - Smallest.

Step 3: Find the Mode

• Count how many times each number appears.
• The number(s) with the highest count is the mode.
• If all numbers appear once, there is no mode.

Step 4: Find the Median

• Count how many numbers there are (n).
• If n is odd: Median = middle number (position = (n+1)/2).
• If n is even: Median = average of the two middle numbers (positions = n/2 and n/2 + 1).

Step 5: Find the Mean

• Add all the numbers together.
• Divide by the total count of numbers.

Step 6: Double-Check

• Mean: Does it make sense? (Should be between the smallest and largest number.)
• Median: Did you order the data correctly?
• Mode: Did you count frequencies properly?
• Range: Did you subtract the right numbers?

Worked Examples

Example 1 – Basic (List of Numbers)

Data: 3, 7, 2, 5, 3, 8

Step 1: Order the data → 2, 3, 3, 5, 7, 8

Step 2: Find the Range → Largest = 8, Smallest = 2 → Range = 8 - 2 = 6

Step 3: Find the Mode → 3 appears twice (most frequent) → Mode = 3

Step 4: Find the Median → 6 numbers (even) → Middle numbers: 3 and 5 (positions 3 and 4) → Median = (3 + 5) / 2 = 4

Step 5: Find the Mean → Sum = 2 + 3 + 3 + 5 + 7 + 8 = 28 → Number of values = 6 → Mean = 28 / 6 ≈ 4.67

What we did and why: - Ordered the data first to make median and mode easier. - Counted frequencies carefully to avoid mode mistakes. - Checked that the mean was between the smallest and largest number.

Example 2 – Medium (Frequency Table – GCSE/A-Level)

Data:

Step 1: Extract the numbers → 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4

Step 2: Find the Range → Largest = 4, Smallest = 1 → Range = 4 - 1 = 3

Step 3: Find the Mode → 2 appears 5 times (most frequent) → Mode = 2

Step 4: Find the Median → Total numbers = 2 + 5 + 3 + 1 = 11 (odd) → Middle position = (11 + 1) / 2 = 6th number → Ordered list: 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4 → Median = 2

Step 5: Find the Mean → Sum = (1×2) + (2×5) + (3×3) + (4×1) = 2 + 10 + 9 + 4 = 25 → Number of values = 11 → Mean = 25 / 11 ≈ 2.27

What we did and why: - Expanded the frequency table into a list to find median and mode. - Used multiplication for the mean to save time (instead of adding all numbers). - Checked that the median position was correct.

Example 3 – Exam-Style (Disguised Data)

Question: A teacher records the number of pets her students have: 0, 1, 2, 1, 3, 0, 4, 1, 2, 1 a) Find the mean, median, mode, and range. b) A new student joins with 10 pets. Which measure changes the most?

Step 1: Order the data → 0, 0, 1, 1, 1, 1, 2, 2, 3, 4

Step 2: Find the Range → Largest = 4, Smallest = 0 → Range = 4 - 0 = 4

Step 3: Find the Mode → 1 appears 4 times (most frequent) → Mode = 1

Step 4: Find the Median → 10 numbers (even) → Middle numbers: 1 and 1 (positions 5 and 6) → Median = (1 + 1) / 2 = 1

Step 5: Find the Mean → Sum = 0 + 0 + 1 + 1 + 1 + 1 + 2 + 2 + 3 + 4 = 15 → Number of values = 10 → Mean = 15 / 10 = 1.5

Part b) New student with 10 pets - New data: 0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 10 - New Range: 10 - 0 = 10 (biggest change) - New Median: 1 (unchanged) - New Mode: 1 (unchanged) - New Mean: (15 + 10) / 11 ≈ 2.27 (increases)

What we did and why: - Ordered the data first to avoid median mistakes. - Noticed that the mean is most affected by outliers (like 10 pets). - Checked which measure changed the most (range increased by 6).

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Right, listen up—this is your last-minute checklist for mean, median, mode, and range. First, order the data—no shortcuts. Range is just biggest minus smallest. Mode is the most common number—count carefully. Median is the middle number, or average of two if even. Mean is total divided by how many numbers. Double-check: Does the mean make sense? Did you order the data? Did you count frequencies right? If the question mentions ‘average,’ it’s probably the mean—but watch for outliers. If it says ‘most common,’ that’s the mode. You’ve got this—go smash it!