Weighted mean (Edexcel GCSE Statistics): Revision Notes
Weighted mean
What is a weighted mean?
A weighted mean is calculated when different data values have varying levels of importance. Instead of treating all values equally, each data value is multiplied by a weight that reflects how significant or important that particular value is to the overall calculation.
This type of mean is particularly useful in situations where some measurements or scores should carry more influence than others in determining the final result.
Weighted means are commonly used in education (exam grades with different weightings), finance (portfolio calculations), and research (survey data with different sample sizes).
The weighted mean formula
The weighted mean is represented by the symbol x̄ (x-bar) and calculated using this formula:
Where:
- w = the weight assigned to each data value
- x = each individual data value
- Σwx = the sum of all (weight × data value) products
- Σw = the sum of all weights
The key difference from a simple mean is that each data value is multiplied by its weight before being summed, and we divide by the sum of weights rather than just the count of values.
How to calculate a weighted mean
Follow these three essential steps to ensure accurate calculation:
- Multiply each data value by its corresponding weight
- Add up all the weighted values (this gives you Σwx)
- Divide this total by the sum of all weights (Σw)
Always double-check your arithmetic at each step, especially when dealing with multiple data points and varying weights.
Worked example 1: Job interview tasks
Imagine you're applying for a job and must complete four tasks: A, B, C, and D. The employer assigns different weights to show which tasks matter most for the role.
| Task | Weight | Jim's mark | Anne's mark |
|---|---|---|---|
| A | 1 | 10 | 3 |
| B | 2 | 8 | 4 |
| C | 2 | 7 | 6 |
| D | 5 | 4 | 8 |
Worked Example: Calculating Job Interview Scores
Jim's weighted mean calculation:
- Step 1: Multiply each mark by its weight:
- Step 2: Add up all the weights:
- Step 3: Divide:
Anne's weighted mean calculation:
- Step 1: Multiply each mark by its weight:
- Step 2: Add up all the weights:
- Step 3: Divide:
Notice how Task D has the highest weight (5), making it the most important factor in determining the final score. Even though Jim scored better on most individual tasks, Anne's strong performance on the most heavily weighted task gave her the higher overall score.
Worked example 2: Flower show competition
At a flower show, displays are judged on three qualities with different levels of importance reflecting what visitors value most.
| Quality | Weight |
|---|---|
| Shape | 1 |
| Colour | 2 |
| Ambience | 2 |
Mr Smith's scores were: Shape = 7, Colour = 9, Ambience = 8
Worked Example: Flower Show Judging
Mr Smith's weighted mean calculation:
- Step 1: Multiply each score by its weight:
- Step 2: Sum of weights:
- Step 3: Weighted mean:
The colour and ambience scores carry twice the importance of the shape score, which is why they have higher weights. This reflects the judging criteria that visual appeal and atmosphere are more critical to the competition than structural form.
Worked example 3: Exam with equal weighting
Sometimes papers are equally weighted, which can simplify calculations while still requiring careful attention to different maximum marks.
Consider an exam with:
- Paper 1: worth 80 marks (Liz scored 52)
- Paper 2: worth 70 marks (Liz scored 56)
- Both papers equally weighted
Worked Example: Equal Weighting Exam
Method 1 - Using percentages:
- Paper 1:
- Paper 2:
- Overall:
Method 2 - Using weights of 1: Since both papers are equally weighted, you can assign each a weight of 1 and add the percentages together, then divide by 2.
This example demonstrates that equal weighting doesn't mean ignoring the weighted mean formula - you still need to account for different maximum marks by converting to percentages first.
Common exam tips and traps
Understanding common mistakes can help you avoid calculation errors and improve your accuracy when working with weighted means.
Key tips for success:
- Always convert marks to percentages first when papers have different total marks
- When weights are equal, you can simply add the percentages and divide by the number of items
- Write down all calculations clearly to avoid errors
- Double-check that your weights add up correctly
Common traps to avoid:
- Forgetting to convert to percentages when papers have different maximum marks
- Adding raw marks instead of weighted marks
- Dividing by the wrong total (using number of items instead of sum of weights)
- Mixing up the order of operations in the formula
Practice problem
Test your understanding with this practical scenario that combines different maximum marks with equal weighting.
An exam has three papers: A, B and C.
- Paper A is worth 60 marks
- Paper B is worth 60 marks
- Paper C is worth 80 marks
- All papers are equally weighted
Ahmed's scores: Paper A = 45, Paper B = 36, Paper C = 60
Your task: Calculate Ahmed's mean percentage.
Hint: Convert each paper to a percentage first, then find the simple average since all papers are equally weighted.
Solution approach:
- Convert each score to a percentage
- Since weights are equal, find the simple average of the percentages
Key Points to Remember:
- Weighted means account for different levels of importance - some data values matter more than others
- The formula is: - multiply each value by its weight, add them up, then divide by the sum of weights
- Always convert to percentages first when dealing with papers or tests that have different maximum marks
- Equal weights mean simple averaging - when all weights are the same, you can just add and divide by the number of items
- Show your working clearly - break down each step to avoid calculation errors