One distant value can change a range without changing median distance

Define the question first

NIST’s guide to measures of scale distinguishes the range, calculated from the largest and smallest values, from median absolute deviation: the median of absolute distances from the data’s median. They summarize different aspects of spread.

The following paper exercise uses original fictional values, not NIST’s example datasets.

Write nine tokens bearing 6, 7, 8, 8, 9, 10, 10, 11 and 12. They have no physical unit, and they do not represent pupils, households, experiments or measurements in any locality.

Our question is deliberately narrow: what happens to two stated calculations when we move one token? A calculator can help, but the sorted list is short enough to check by hand.

Keep repeated values as separate observations. There really are two eights and two tens in this invented collection.

Find the middle, then the distances

The list is already in increasing order. With nine tokens, the fifth position is the middle: four tokens lie before it and four after it.

That fifth value is 9. We use this odd-sized list so that finding its median does not require choosing a rule for averaging two middle positions.

Write a separate row measuring each token’s distance from 9. A token at 6 is three away; a token at 12 is also three away.

Direction does not matter in this distance row.

Following the original order, the nine distances are 3, 2, 1, 1, 0, 1, 1, 2 and 3. Sort those distances themselves: 0, 1, 1, 1, 1, 2, 2, 3 and 3.

Their fifth value is 1. This is our unscaled median absolute deviation, abbreviated MAD.

The second sorting step matters. Reading the fifth entry of the unsorted distance row would give zero, but zero is not its median.

Nor should we simply take the median of the original positive tokens and call 9 a distance.

Calculate the endpoints separately

For the same list, the range is 12 minus 6, giving 6. Record both results: range 6 and MAD 1.

Neither result says that every token is one unit from the center or that the range contains six observations. The first calculation subtracts two endpoint values; the second locates the middle value in an entirely different list of nine distances.

Keeping the intermediate rows visible is more useful than copying two unlabeled answers into a worksheet.

For an extra check, add the original distance row: its total is 14. Dividing by nine gives about 1.556.

That is the arithmetic average of our distances from 9, not their median of 1. We are defining this extra worksheet calculation explicitly, not assigning it another statistical abbreviation.

It catches a common transcription mistake: a spreadsheet’s average command and its median command perform different operations. The fact that both results describe the same distance row does not make them interchangeable.

Move only the last token

Replace the final token, 12, with 52. The new ordered list is 6, 7, 8, 8, 9, 10, 10, 11 and 52.

Nothing else changes. The median remains 9, because the fifth position still contains 9.

The new distances from it are 3, 2, 1, 1, 0, 1, 1, 2 and 43. After sorting, they become 0, 1, 1, 1, 1, 2, 2, 3 and 43.

The fifth distance is still 1, so this worksheet’s MAD has not changed.

The range, however, becomes 52 minus 6, or 46. Our additional average-distance check becomes 54 divided by nine, or 6.

We now have three transparent results from the altered list: range 46, median distance 1 and average distance from 9 equal to 6. This is not a contradiction.

The moved token enters the endpoint subtraction directly, contributes 43 to the distance sum, and remains above the fifth position in the sorted distance list. Each calculation uses it differently.

Compare a different middle arrangement

Now put the altered list aside and start a second invented collection: 1, 3, 5, 7, 9, 11, 13, 15 and 17. Again the median is 9.

Its distances from 9, in order, are 8, 6, 4, 2, 0, 2, 4, 6 and 8. Sorted, they are 0, 2, 2, 4, 4, 6, 6, 8 and 8.

The fifth distance is 4. Its range is 17 minus 1, or 16.

Compare that second collection with the altered first collection. The second has a smaller endpoint range, 16 rather than 46, but a larger median distance, 4 rather than 1.

We cannot answer the question ‘Which is more spread out?’ without saying which calculation we mean.

This is the worksheet’s central comparison, not a ranking of real schools, services or scientific instruments. The exact lists explain the apparent reversal: one holds several tokens near 9 plus a distant endpoint, while the other places more tokens farther from 9.

Check translation and scale

Return to the original first collection. Add 100 to every token.

Its middle becomes 109, and its endpoints become 106 and 112. Each distance from the new middle matches its old distance from 9, so MAD stays 1 and range stays 6.

Alternatively, multiply every original token by three. The median becomes 27, the endpoints become 18 and 36, MAD becomes 3 and range becomes 18.

These two changes test whether we are tracking positions and distances consistently.

Keep all three original lists, the altered token and both distance rows beside the final summaries. Do not delete the 52 merely because one summary did not change; we invented it intentionally, and it remains part of that version of the exercise.

Real unusual observations require their own evidence, not a decision copied from fictional arithmetic. Our result is limited but useful: a number describing spread needs a named calculation, visible inputs and a clear question.

A stable middle-distance summary does not establish that nothing important happened at the edge.