How Quantities Combine: Understanding Data

When groups of different sizes are combined, the result is a weighted average. The same idea gives mixtures, mark schemes and stacked bar charts.

Key ideas

A combined average is not the simple average of the group averages unless the groups are the same size.
The combined average always lies between the group averages, closer to the bigger group.
In a mixture, the price or strength of the mix is a weighted average of the parts.
Alligation gives the ratio in which two things must be mixed to reach a target value.
With negative marking, wrong answers subtract marks, so the score can be less than the number correct times the mark.
A stacked bar shows amounts; a 100% stacked bar shows shares. A bigger share does not always mean a bigger amount.

Remember

Weighted average	(w₁x₁ + w₂x₂ + …) ÷ (w₁ + w₂ + …)
Two groups combined	(n₁m₁ + n₂m₂) ÷ (n₁ + n₂)
Alligation (cheaper c, dearer d, mix m)	cheaper : dearer = (d − m) : (m − c)
Share in a 100% bar	part ÷ total × 100%

Watch out

Averaging the two averages when the groups are of different sizes.
Forgetting to subtract the marks for wrong answers.
Reading a 100% stacked bar as if it showed actual numbers.

Worked example

A school counts term 1 as 25%, term 2 as 25% and the final exam as 50%. Rahul scores 96, 82 and 46 (out of 100). Find the overall mark, and say how much higher or lower it is than the simple average.

Working: 25% of 96 + 25% of 82 + 50% of 46 = 67.5; simple average = 224 ÷ 3 = 74.67

Answer: 67.5; 7.17 lower than the simple average