Why is the range of 3, 8, 5, 12, 7 equal to 9 and not 7?

Because the range is the largest value minus the smallest value: 12 − 3 = 9. It measures how spread out the data is. The answer 7 is the mean (3 + 8 + 5 + 12 + 7 = 35, then 35 ÷ 5 = 7) — a different measure that answers a different question. Range and mean are not interchangeable.

What is the difference between an outlier and the mode?

An outlier is a value that sits far away from all the others. The mode is the value that appears most often. They are different measures. For the heights 98, 103, 91, 85, 159, 102, 91, the outlier is 159 (far above the 85–103 cluster), while 91 is the mode (it appears twice). A value being repeated does not make it the outlier.

How do I know which measure a worded question is asking for?

Match the wording to the definition. 'Most common' or 'most frequent' means the mode. 'Middle value' means the median. 'Average' usually means the mean (total ÷ how many). 'Spread' or 'largest minus smallest' means the range. 'Far from the rest' means an outlier. Read the named measure first, then compute only that one.

GCSE Maths Foundation · AQA · Averages & spread

Confusing statistical measures: range vs mean, outlier vs mode

Mean, median, mode, range and outlier are not five names for the same thing. Students who treat them as interchangeable labels for “the answer” compute the mean when the question asks for the range, or give the repeated value (the mode) when it asks for the outlier. Each measure has its own definition and its own answer on the very same data.

The thirty-second fix: read the measure the question names before you calculate — the range is largest − smallest, the mean is total ÷ how many, the mode is the most frequent value, and an outlier is a value far from the rest.

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How to spot it in your own work

You gave $7$ as the range of 3, 8, 5, 12, 7 — but 7 is the mean ( $35 \div 5$ ), not the range ( $12 - 3 = 9$ ).
You named the repeated value as the outlier — but a repeated value is the mode; the outlier is the value far from the rest.
You computed an average without checking which one the question named (mean, median or mode).
You answered with the largest value instead of the range (largest minus the smallest).

An exam question that triggers it

Here is the canonical AQA Foundation trigger (Nov24 P2 Q5a shape):

Work out the range of 3, 8, 5, 12, 7.

The misconception answer is $7$ — the mean of the five values ( $35 \div 5$ ). But the range is a measure of spread: largest minus smallest.

Largest = $12$ , smallest = $3$ , so the range is $12 - 3 = 9$ .

Why students fall for this

The named measures all live in one slot in the student’s head — “the statistics answer” — so the specific word in the question (range, mean, mode, outlier) is not read as a precise instruction. The student reaches for whichever calculation is most familiar, usually the mean, regardless of what was asked.

The outlier-versus-mode muddle has the same root. A value like $91$ “stands out” because it is repeated, so the student offers it as the outlier — but “stands out” as repeated is the mode, while “stands out” as far from the rest is the outlier. Two different ideas collapse into one.

The fix is not more arithmetic — it is holding the identity of each measure. Once the student computes all of mean, median, mode and range on a single data set and sees the answers differ, the labels stop being swappable.

The fix: Read the measure first, then compute only that one

Range = largest − smallest. For $3, 8, 5, 12, 7$ : largest 12, smallest 3, so range = $12 - 3 = 9$ . The mean ( $35 \div 5 = 7$ ) is a different measure.

Mode = most frequent value. For $4, 7, 4, 9, 6$ : 4 appears twice, so the mode is $4$ . The range of the same set is $9 - 4 = 5$ — not equal to the mode.

Outlier = value far from the rest. For $98, 103, 91, 85, 159, 102, 91$ : the cluster runs 85–103, so the outlier is $159$ . The repeated value $91$ is the mode, not the outlier.

Worked example

Find the mean, median, mode and range of $4, 7, 4, 9, 6$ .

Sort the data.
$4, \; 4, \; 6, \; 7, \; 9$
Mean = total ÷ how many = $30 \div 5 = 6$ .
Median = middle of the sorted list = $6$ .
Mode = most frequent value = $4$ .
Range = largest − smallest = $9 - 4 = 5$ .

Four measures, four definitions: mean 6, median 6, mode 4, range 5. They do not all agree — which is exactly why you cannot swap one name for another.

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Common questions

Is the range the same as the mean?: No. The range is the largest value minus the smallest (a measure of spread); the mean is the total divided by how many (a measure of a typical value). For $3, 8, 5, 12, 7$ the range is $12 - 3 = 9$ and the mean is $35 \div 5 = 7$ — different numbers, different measures.
How do I tell an outlier from the mode?: An outlier is a value far away from all the others; the mode is the value that appears most often. A repeated value is the mode, not necessarily the outlier. In $98, 103, 91, 85, 159, 102, 91$ , the outlier is $159$ (far from the rest) and the mode is $91$ (the repeated value).
Which measure does “most common” mean?: “Most common” or “most frequent” names the mode — the value that occurs most often. For shoe sizes $6, 7, 7, 8, 7, 9$ , the size 7 appears three times, so the mode is $7$ .
Why do all four measures give different answers on the same data?: Because each measures a different feature of the data. The mean and median describe a typical value, the mode describes the most frequent value, and the range describes the spread. On $4, 7, 4, 9, 6$ they are 6, 6, 4 and 5 — the fact that they differ is the proof that the names are not interchangeable.

Related misconceptions

Computing mean and median: why you must sort before finding the middleThe companion misconception: once you know which measure is named, this is how you actually compute the mean and the median correctly.
Fraction of an amount: why 2/5 of 1020 is 408, not 40A related label-versus-operation error: reading a word as a slot to fill rather than an instruction to carry out.
Function machines: why you must invert each step to find the inputAnother case where reading the question precisely — which operation, in which direction — is what protects the answer.

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