Weighted averages rarely look difficult. A passage describes two groups, gives an average for each, and asks about the whole. The arithmetic involved is multiplication, addition and one division, which nobody preparing for a master's degree finds hard. The trouble is that the wrong method is also easy and produces a very plausible number, so the mistake does not feel like a mistake. The syllabus overview lists weighted averages among the calculations the General Academic Module can draw on, alongside percentages, ratios and rates. This post explains where the slip happens, how to set the calculation up so it cannot happen, and two quick checks to run before you choose an option.
Why the average of averages misleads
Suppose a passage reports a study with two groups. Group A has 20 participants with an average score of 60. Group B has 80 participants with an average score of 80. What is the average across all 100 participants?
The tempting answer is 70, the midpoint of 60 and 80. That would only be right if the two groups were the same size. Here, four out of every five participants are in Group B, so the overall average must sit much closer to 80 than to 60.
The correct calculation goes back to totals. Group A contributes 20 × 60 = 1,200 points. Group B contributes 80 × 80 = 6,400. Together that is 7,600 points across 100 people, so the overall average is 76.
The same trap appears with percentages, and there it is even easier to miss. If 40% of 1,000 people in one region say yes and 10% of 4,000 people in another region say yes, the combined figure is not 25%. That is 400 plus 400 yes answers out of 5,000 people, or 16%. A percentage is an average too, one that happens to be written with a % sign, and it has to be weighted by the size of the group it describes. Our guide to percentages and ratios covers the related habit of asking "percentage of what?" before comparing two figures.
Setting the calculation up so it cannot go wrong
The reliable method has three steps, and it works whether the passage gives counts, percentages or rates.
- Turn each average back into a total. Multiply each group's average by its size. If the passage gives shares rather than counts (say, 30% of the sample is in one group), use the shares directly as weights.
- Add the totals and add the sizes. Keep the two sums apart. Mixing them is where most arithmetic slips come from.
- Divide once, at the end. One division, of total by total, gives the combined average.
Before you start, check what the passage actually provides. Sometimes the sizes come from a table and the averages from the text, and the two may not describe exactly the same groups. If a table splits a sample by age and the text gives averages by region, you cannot combine them. The question will usually be asking about something else. Our note on interpreting graphs and tables covers confirming that two sources share the same categories before you merge them.
A faster route when the numbers allow it
Once the idea is familiar, there is a shortcut that often removes the need for totals altogether. The combined average always lies between the two group averages, and it sits closer to the larger group in proportion to the sizes.
In the example above, the groups are 20 and 80, a ratio of 1 to 4. The gap between the averages is 20 points. The combined average therefore lies one fifth of that gap away from the larger group's average: 80 minus 4 equals 76. The answer is the same, and with friendly numbers it takes a few seconds.
You can also run the method in reverse, and passages sometimes ask for exactly that. If you know the overall average and one group's average, you can find the other. A hundred people average 76. Twenty of them average 60. The remaining 80 must account for 7,600 minus 1,200 = 6,400 points, which is an average of 80. Thinking in totals makes the reverse question no harder than the forward one.
Prepare to do this without a calculator. The official exam rules for your sitting are the place to confirm what is permitted. Either way, the numbers in a well-designed question tend to be chosen so that a clean mental route exists, and estimation is often enough to rule out most of the options.
When the mix changes, not the groups
The most interesting weighted-average questions are not really about arithmetic. They are about what can happen to an overall figure when the make-up of the groups shifts.
Imagine a passage comparing two years. In the first year, half of a sample of 100 is in Group A, averaging 60, and half is in Group B, averaging 80, so the overall average is 70. In the second year, both groups improve by two points, to 62 and 82. But now 90 people are in Group A and only 10 in Group B. The overall average is (90 × 62 + 10 × 82) ÷ 100 = 6,400 ÷ 100 = 64.
Every group improved, yet the overall figure fell by six points. Nothing is wrong with the data. The weights moved towards the lower-scoring group. A passage might present the falling overall figure as evidence that something got worse, and a question might ask whether that conclusion follows. It does not, at least not without looking at the groups separately.
When a passage reports a change in an overall average or rate, ask two questions. Did the groups themselves change, or did the proportions change? And does the conclusion in the passage depend on which of those happened?
Two checks before you answer
- The range check. A combined average must lie between the smallest and largest group averages. An option outside that range is wrong, however neat it looks.
- The direction check. The combined figure must lean towards the larger group. If your answer sits closer to the smaller group's average, you have probably swapped the weights or averaged the averages.
Both take a moment and catch nearly every version of the slip.
Practising it
Weighted averages reward a small number of repetitions done deliberately. Take a few published tables that report results by group, combine two rows in your head using totals, then confirm with the proportion shortcut. After a handful of attempts the "average of averages" answer starts to look suspicious on sight, which is the habit you want on test day. To see how this sits alongside the other passage-based tasks, try the free dMAT diagnostic.
