Weighted Averages and Indices

Weighted averages and index numbers occasionally appear in UCAT DM data interpretation questions. They test whether you understand that combining data from groups of different sizes requires weighting, and whether you can spot the dangers of aggregating data inappropriately. This lesson also introduces Simpson's paradox — a counter-intuitive phenomenon where a trend that appears in subgroups reverses when the groups are combined.

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Simple Average vs Weighted Average

Simple (Unweighted) Average

$\text{Simple average} = \frac{\text{Sum of all values}}{\text{Number of values}}$Simple average=Number of valuesSum of all values​

A simple average treats every value equally. This is appropriate when all groups are the same size.

Weighted Average

$\text{Weighted average} = \frac{\sum (\text{value} \times \text{weight})}{\sum \text{weights}}$Weighted average=∑weights∑(value×weight)​

A weighted average accounts for the size or importance of each group.

When to Use Each

Situation	Use
Same-sized groups	Simple average is fine
Different-sized groups	Weighted average required
Individual data points	Simple average is fine
Group summaries with different Ns	Weighted average required

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Worked Example: Weighted Average