Given some data with corresponding weights, how would the weighted average change if all the weights are doubled? If required, experiment with some data. What do you observe? Justify your answer using algebra.

Step-by-step solution

Idea: Only the ratio of the weights matters. Weights 3 : 2 : 5 and 6 : 4 : 10 are the same ratio.

1. Experiment: marks 60, 64, 73 with weights 3, 2, 5 give 180 + 128 + 36510 = 67.3. With weights 6, 4, 10: 360 + 256 + 73020 = 134620 = 67.3. Same.1 mark
2. Algebra: new average = (2w₁)x₁ + (2w₂)x₂ + … + (2w_n)x_n2w₁ + 2w₂ + … + 2w_n = 2(w₁x₁ + … + w_nx_n)2(w₁ + … + w_n).1 mark
3. Cancel the common factor 2: it equals the original weighted average. Observation: doubling (indeed, multiplying all weights by any positive number) does not change the weighted average; only the ratio of the weights matters.1 mark

Answer to write in the exam

Common mistakes that cost marks

• Thinking the average doubles. The denominator doubles too.
• Doubling the weights in the numerator only.
• Confusing this with adding the same number to every weight, which does change the average.

How this can come in the exam

Try one yourself

Find the weighted mean of 4, 8, 10 with weights 1, 1, 2 and then with weights 5, 5, 10.

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