Range, interquartile range and standard deviation: what moves and what does not

6 min read·Updated 5 September 2026

Measures of dispersion answer one question — how spread out is this data? — with three different numbers, each sensitive to something different. Range sees only the two extremes. Interquartile range sees only the middle half, and is therefore blind to outliers by design. Standard deviation sees every value, and weights distance from the mean quadratically.

Paper 2 rarely asks you to compute these from scratch. It asks what happens to them when the data changes.

Misconception 1: swapping range for interquartile range

Both are called "spread" and both are a subtraction, so they get substituted for one another freely. They are not interchangeable, and the whole point of having two is that they disagree: adding one extreme value to a data set can change the range enormously and leave the interquartile range exactly as it was.

When a question says a new value was added and asks which measures changed, that asymmetry is the question. Work out each one separately; do not reason about "the spread".

Misconception 2: getting transformations backwards

Add the same constant to every value and the data slides along the number line without stretching: the range, the interquartile range and the standard deviation are all unchanged. The mean and the median move by that constant.

Multiply every value by a positive constant and the data stretches: all three measures of spread are multiplied by that constant. (Variance, being a square, is multiplied by its square.)

The intuition that trips people is arithmetic's: adding feels like the operation that "changes things". For spread it is the one that does not.

Misconception 3: reading a "greatest possible" spread off the data shown

As with mean and median, a question asking for the largest possible range under some constraint is asking you to construct a data set, not to measure the one in front of you. Push the extremes apart as far as the constraints allow, then re-check every constraint at that extreme.

A worked example

Consider the data set

The mean is $40 \div 5 = 8$, and the deviations from it are $-4, -1, -1, 1, 5$. Squaring and averaging gives the variance

so the standard deviation is $\sqrt{8.8} \approx 2.97$. The range is $13 - 4 = 9$.

Now add $5$ to every value, giving $9, 12, 12, 14, 18$. The mean becomes $13$, but the deviations are still $-4, -1, -1, 1, 5$ — every value moved and so did the mean, so nothing about the distances changed. The standard deviation is still $2.97$ and the range is still $9$.

Instead multiply the original values by $2$, giving $8, 14, 14, 18, 26$. The mean becomes $16$ and the deviations double to $-8, -2, -2, 2, 10$. The variance becomes $176 \div 5 = 35.2$, which is $4$ times the original, and the standard deviation becomes $\approx 5.93$ — exactly double. The range doubles to $18$.

The wrong turn is to expect the $+5$ to widen the spread and the $\times 2$ to leave it alone. Both errors survive a sanity check on the mean, because the mean genuinely does change in both cases.

Check yourself

One line: "shifting moves the centre, stretching moves the spread." And when a question adds or removes a single value, compute range and interquartile range separately — the answer usually depends on them behaving differently.

The drill below generates new data sets and transformations each time, so the rule gets exercised rather than recited.

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