Given some data with corresponding weights, what would happen to the weighted average if all the weights are increased by a constant value, say 1? If required, experiment with some data. What do you observe? Justify your answer using algebra.

Step-by-step solution

Idea: Adding 1 to every weight adds one extra copy of every value, which is like mixing the old weighted data with the plain data. The result is a weighted mean of the old weighted average and the plain mean.

1. Experiment: values 10, 20 with weights 1, 3: 10 + 604 = 17.5. Weights 2, 4: 20 + 806 ≈ 16.67. The plain mean is 15, so the average moved from 17.5 towards 15.1 mark
2. Algebra: let W = w₁ + … + w_n, the weighted average A = Σw_ix_iW and the plain mean m = Σx_in. New average = Σ(w_i + 1)x_iΣ(w_i + 1) = Σw_ix_i + Σx_iW + n.1 mark
3. Since Σw_ix_i = WA and Σx_i = nm, the new average = WA + nmW + n: a weighted mean of A (weight W) and m (weight n). So it lies between A and m.1 mark
4. Observation: adding 1 to all weights pulls the weighted average towards the plain mean, making the weights less important. It is unchanged only when A = m (e.g. equal weights). Unlike doubling, adding a constant changes the ratio of the weights (1 : 3 became 2 : 4 = 1 : 2).1 mark

Check: Experiment: W = 4, A = 17.5, n = 2, m = 15: 4 × 17.5 + 2 × 156 = 1006 ≈ 16.67 ✓, the same as computed directly.

Answer to write in the exam

Common mistakes that cost marks

• Assuming it stays the same, as it does when weights are doubled. Adding a constant changes the ratio of the weights.
• Adding 1 to the weights in the denominator only (or in the numerator only).
• Concluding from one example without the algebra.

How this can come in the exam

Try one yourself

Values 6 and 12 have weights 3 and 1. Find the weighted average, then find it again after adding 1 to each weight.

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