A list of numbers can answer several different questions. You might ask what equal share its total would give, which value sits in the middle, or how far its extremes are apart. Those questions lead to the arithmetic mean, median and range. Start by identifying which question the requested summary is intended to answer.
These examples use invented lists, so they do not report results from a real survey. Keep every observation in its place, including repeated values if a later question contains them. Avoid choosing whichever calculation is most familiar: the definition in the question determines what to do with the list.
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Use a total for the arithmetic mean
For 2, 4 and 9, the total is 15 and there are three observations. The arithmetic mean is therefore 15 ÷ 3 = 5. You can interpret this as an equal share of the original total. It need not be one of the values that appeared in the list.
Count observations separately from adding their values. A denominator of three comes from the number of entries, not from the number of different digits or the largest entry. After calculating, multiplying the mean by the observation count reconstructs this total.
Use position for the median and endpoints for the range
The ordered list 1, 3, 8, 10, 20 contains five values. Two lie before 8 and two after it, so 8 is the median. Adding all five values would address the mean instead. Ordering is part of the procedure, even when the supplied list is already ordered.
For the list 4, 7, 9, 15, the question defines range as maximum minus minimum. That gives 15 − 4 = 11. The middle entries still belong to the list, but they do not enter this endpoint subtraction. Compare the kind of information each summary retains before treating summaries as interchangeable.
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Check your understanding with examples
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Example 1 · Probability and statistics
What is the arithmetic mean of 2, 4 and 9?
5
4
9
15
Read the answer and explanation
Answer: 5
Add the three values and divide by their number: (2 + 4 + 9)/3 = 15/3 = 5.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Example 2 · Probability and statistics
What is the median of 1, 3, 8, 10 and 20?
8
3
10
8.4
Read the answer and explanation
Answer: 8
The five values are already ordered. The middle value is the third, 8. A median uses position rather than the sum of values.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Example 3 · Probability and statistics
What is the range, defined as maximum minus minimum, of 4, 7, 9 and 15?
4
35
11
15
Read the answer and explanation
Answer: 11
The largest value is 15 and the smallest is 4. Their difference is 15 − 4 = 11.
Per-ID original mathematical derivations and faithful English adaptations for 300 existing families, plus original fictional data and diagram conditions. Supports mathematical reasoning, not external facts or independent review approval.
Source checked: 2026-10-04 · Original authoring and derivation record
Topics: Probability and statistics. Range: Everyday knowledge, Broader knowledge. Difficulty: Basic, Standard. These are selected initially. You can change these on the setup screen.
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