Mean, median and mode: when the question asks for the largest possible one

6 min read·Updated 5 September 2026

Computing a mean, a median or a mode from a list of numbers is not what this topic tests by the time it reaches Paper 2. The interesting questions describe a data set only partly — a number of values, a mean, a constraint on the mode — and ask what some other measure could be. That turns a calculation into a small piece of reasoning, and it is the switch between those two modes of thinking that decides the mark.

Misconception 1: assuming the mode is unique

The mode is the most frequent value, and nothing forces exactly one value to win. A set can have two modes, three, or none at all when every value appears equally often. Students who have only ever met sets with a single clear mode read "the mode" as a promise that one exists and is unique, and pick whichever candidate comes first in the list.

This matters most in constraint questions, where "the mode is 4" and "4 is a mode" are genuinely different conditions, and the difference changes the answer. Read which one the question stated.

Misconception 2: answering "greatest possible" from the given arrangement

When a question says greatest possible median or least possible mean, it is not describing one data set. It is describing a family of data sets, and asking you to find the extreme member. Reading a value off the arrangement in front of you answers a question nobody asked.

The move is always the same: decide which direction each free value must be pushed to make the target extreme, then check that the constraints still hold at that extreme. Very often the constraint you forget to re-check is the one that makes the answer one step less extreme than you first thought.

Misconception 3: assuming the part resembles the whole

A subset of a data set need not have a median anywhere near the full set's median, and adding a value to a set can move the mean past values that were already in it. "The middle of a part is near the middle of the whole" is an intuition about symmetric data, and the data in these questions is chosen precisely because it is not symmetric.

A worked example

Five positive integers have a mean of $6$, and their only mode is $4$. Find the greatest possible median.

The mean fixes the total: $5 \times 6 = 30$. To push the median (the third value once sorted) as high as possible, the two smallest values should be as small as the constraints allow — and the mode gives them to us, since $4$ must appear more often than any other value. Take the two smallest to be $4, 4$, leaving three values summing to $22$, each at least $4$.

The tempting next step is to make the median as large as arithmetic allows: $22 \div 3 \approx 7.3$ suggests trying $7$, giving $4, 4, 7, 7, 8$. That fails, and it fails on the constraint that is easy to forget — $7$ now appears twice as well, so $4$ is no longer the only mode. Step down to $6$:

So the greatest possible median is $6$. The wrong answer $7$ is not a slip — it is what you get by optimising and then not re-checking the condition you optimised against.

Check yourself

When a question contains the word possible, ask "which values am I free to choose, and which way do I push them?" — then, before writing the answer, re-read every condition and confirm the extreme arrangement still satisfies all of them.

The drill below generates new constraint sets each time, so the reasoning is the part you practise.

Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.