How Quantities Combine: Understanding Data
The average of averages
Find the average of two groups put together, and see why group sizes matter.
A badminton academy has 8 seniors averaging 165.5 cm and 3 juniors averaging 149.33 cm. "Is the whole group's average (165.5 + 149.33)/2 = 157.415?" No.
The true average adds all 11 heights and divides by 11: 1772/11 ≈ 161.09 cm.
Why the shortcut fails: averaging the two averages splits the vote 50–50, as if the 3 juniors counted as much as the 8 seniors. They don't: there are far more seniors, so the answer should sit nearer the seniors' 165.5. Predict which side the answer falls before you calculate.
Use totals: total = average × count.
- seniors: 165.5 × 8 = 1324;
- juniors: 149.33 × 3 ≈ 448;
- combined average = (1324 + 448)/11.
In general, groups of sizes n and m with averages a and b combine to (an + bm)/(n + m).
When the shortcut does work. Jaspreet cycled 5 days in each of 3 weeks. Every week has the same size, so averaging the weekly averages is fine: (5a + 5b + 5c)/15 = (a + b + c)/3.
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