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

Two hospital sites report patient satisfaction scores. Site A has 200 patients and an average score of 85. Site B has 800 patients and an average score of 72. What is the overall average satisfaction score?

Wrong (simple average): (85 + 72) / 2 = 78.5

Correct (weighted average):

$\frac{(200 \times 85) + (800 \times 72)}{200 + 800} = \frac{17,000 + 57,600}{1,000} = \frac{74,600}{1,000} = \textbf{74.6}$200+800(200×85)+(800×72)​=1,00017,000+57,600​=1,00074,600​=74.6

The correct answer (74.6) is much closer to Site B's score (72) than to Site A's score (85), because Site B has four times as many patients.

UCAT Tip: Whenever you see averages from groups of different sizes, you MUST use the weighted average. The simple average of the group averages is almost always wrong.

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Simpson's Paradox

Simpson's paradox occurs when a trend observed in subgroups reverses when the data is combined. It is one of the most important concepts in statistical reasoning, and it occasionally appears in UCAT DM questions.

Classic Example: University Admissions

	Department A	Department B
Male applicants	50 applied, 40 admitted (80%)	400 applied, 200 admitted (50%)
Female applicants	400 applied, 340 admitted (85%)	50 applied, 30 admitted (60%)

Within each department, women have a HIGHER admission rate:

• Dept A: Women 85% vs Men 80%
• Dept B: Women 60% vs Men 50%

But overall:

• Men: (40 + 200) / (50 + 400) = 240 / 450 = 53.3%
• Women: (340 + 30) / (400 + 50) = 370 / 450 = 82.2%

Wait — in this example, women have a higher overall rate too. Let me adjust the numbers to show the paradox properly.

	Department A (hard to get in)	Department B (easy to get in)
Male applicants	10 applied, 1 admitted (10%)	90 applied, 72 admitted (80%)
Female applicants	80 applied, 16 admitted (20%)	20 applied, 18 admitted (90%)

Within each department, women have a HIGHER admission rate:

• Dept A: Women 20% vs Men 10%
• Dept B: Women 90% vs Men 80%

But overall:

• Men: (1 + 72) / (10 + 90) = 73 / 100 = 73%
• Women: (16 + 18) / (80 + 20) = 34 / 100 = 34%