GCSE Maths Statistics and Probability - 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 (Physics, Chemistry, Biology) – Complete Guide

Introduction

"Mastering mean, median, mode, and range doesn’t just get you marks—it unlocks real-world data analysis in medicine, sports science, and environmental studies. In your GCSE/A-Level exams, this topic appears in at least 2-3 questions per paper, often worth 4-6 marks total. Miss it, and you’re leaving easy marks on the table."

WHAT YOU NEED TO KNOW FIRST

Before diving in, ensure you understand: 1. Ordering numbers – How to arrange data from smallest to largest. 2. Basic arithmetic – Addition, division, and subtraction. 3. Frequency tables – How to read and interpret data in tables.

KEY TERMS & FORMULAS

Key Terms

Formulas

1. Mean
   [
   \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
   ]

2. MEMORISE THIS – Not always given in exams.

3. Median

4. If odd number of values: Middle number.
5. If even number of values: Mean of the two middle numbers.

6. MEMORISE THIS – Examiners expect you to know this.

7. Mode

8. The number that appears most often.

9. No formula – Just count frequencies.

10. Range
    [
    \text{Range} = \text{Maximum value} - \text{Minimum value}
    ]

11. MEMORISE THIS – Simple but often forgotten.

STEP-BY-STEP METHOD

Step 1: Organise the Data

• If given a list, order it from smallest to largest.
• If given a table, extract the numbers into a list.

Step 2: Find the Mean

1. Add all the numbers together.
2. Divide by the total number of values.

Step 3: Find the Median

1. Count how many numbers there are.
2. If odd, the median is the middle number.
3. If even, add the two middle numbers and divide by 2.

Step 4: Find the Mode

1. Count how many times each number appears.
2. The number with the highest frequency is the mode.
3. If no repeats, write: "No mode."
4. If multiple numbers have the same highest frequency, list them all.

Step 5: Find the Range

1. Identify the largest and smallest numbers.
2. Subtract the smallest from the largest.

WORKED EXAMPLES

Example 1 – Basic List

Data: 5, 2, 8, 2, 6, 1, 4

Step 1: Order the Data

1, 2, 2, 4, 5, 6, 8

Step 2: Mean

Sum = 1 + 2 + 2 + 4 + 5 + 6 + 8 = 28 Number of values = 7 Mean = 28 ÷ 7 = 4

Step 3: Median

7 numbers → odd → 4th number = 4

Step 4: Mode

"2" appears twice (most frequent) → 2

Step 5: Range

Largest = 8, Smallest = 1 Range = 8 – 1 = 7

Answer: Mean = 4, Median = 4, Mode = 2, Range = 7

What we did and why: - Ordered data first to make median and mode easier. - Mean required adding all numbers before dividing. - Mode was found by counting frequencies.

Example 2 – Medium (Frequency Table)

Data:

Step 1: Extract Numbers

3, 3, 5, 5, 5, 5, 7, 7, 7

Step 2: Mean

Sum = (3×2) + (5×4) + (7×3) = 6 + 20 + 21 = 47 Number of values = 2 + 4 + 3 = 9 Mean = 47 ÷ 9 ≈ 5.22

Step 3: Median

9 numbers → odd → 5th number = 5

Step 4: Mode

"5" appears 4 times (most frequent) → 5

Step 5: Range

Largest = 7, Smallest = 3 Range = 7 – 3 = 4

Answer: Mean ≈ 5.22, Median = 5, Mode = 5, Range = 4

What we did and why: - Used frequency to expand the list. - Mean required multiplying each number by its frequency. - Mode was clear because one number had the highest frequency.

Example 3 – Exam-Style (Disguised Data)

Question: A biologist records the number of eggs in 10 bird nests: 3, 5, 2, 6, 3, 4, 5, 2, 6, 4 Calculate the mean, median, mode, and range.

Step 1: Order the Data

2, 2, 3, 3, 4, 4, 5, 5, 6, 6

Step 2: Mean

Sum = 2+2+3+3+4+4+5+5+6+6 = 40 Number of values = 10 Mean = 40 ÷ 10 = 4

Step 3: Median

10 numbers → even → Mean of 5th and 6th numbers = (4 + 4) ÷ 2 = 4

Step 4: Mode

"2, 3, 4, 5, 6" all appear twice → No unique mode (or list all: 2, 3, 4, 5, 6)

Step 5: Range

Largest = 6, Smallest = 2 Range = 6 – 2 = 4

Answer: Mean = 4, Median = 4, Mode = No unique mode (or 2, 3, 4, 5, 6), Range = 4

What we did and why: - Ordered data to find median easily. - Mean required careful addition. - Mode had multiple answers—examiners expect you to recognise this.

COMMON MISTAKES

EXAM TRAPS

1-MINUTE RECAP

"Night before the exam? Here’s what you need to remember: 1. Mean = Sum ÷ Number of values. Add them all, then divide. 2. Median = Middle number. Order first, then find the middle. 3. Mode = Most frequent. Count how many times each number appears. 4. Range = Biggest – Smallest. Simple subtraction. Common traps? Forgetting to order data, miscounting for mean, and not checking for multiple modes. Double-check your work—examiners love to test these! You’ve got this!"