Which measures of central tendency get affected if the extreme observations on…

In any sorted (ascending or descending) data set, the median is fixed by rank — it is simply the middle value (or the average of the two middle values) — while the mean is computed from the sum of every value divided by the count, and the mode is whichever value occurs most often. Whether trimming values from the two ends of the data changes a measure depends on whether that measure is driven by rank position or by the actual values removed.

Here, one extreme observation is removed from each end of the sorted data. With n observations to start, dropping exactly one from the top and one from the bottom leaves n minus 2 observations, and the following happens to each measure:

1. Median: because exactly one value is dropped from each side, the middle rank position(s) of the remaining data line up with the same middle position(s) as before — the median value does not move.

2. Mean: the sum of the data decreases by the two removed extreme values and the count decreases by 2. There is no positional rule protecting this ratio the way there is for the median, so removing two specific values generally changes the mean — it would take a coincidental cancellation for the ratio to land back on its original value, so mean is properly treated as an affected measure.

3. Mode: the most frequent value's status depends on how many times it occurs and whether any of those occurrences sit at the extremes being removed. If none of them do, its frequency is untouched and it remains the mode; if one of them does, that removal reduces its frequency by one, which may or may not be enough to cost it the lead — it depends on how the remaining counts compare. So mode carries no guarantee of staying fixed the way median does, but it is not automatically dislodged just because one of its occurrences happens to be an extreme, either.

Check with two small examples. First, sorted data 2, 4, 4, 6, 9: mean = 25/5 = 5, median = 4 (middle of five values), mode = 4 (appears twice, more than any other value). Removing the extremes 2 and 9 leaves 4, 4, 6: mean = 14/3 (approximately 4.67, changed), median = 4 (still the middle value — unchanged, confirming the rank argument), and mode stays 4 because neither of its two occurrences was an extreme being removed.

Second, sorted data 2, 2, 4, 6, 9: mean = 23/5 = 4.6, median = 4 (middle of five values), mode = 2 (appears twice, more than any other value, and one of its occurrences is the lower extreme). Removing the extremes 2 and 9 leaves 2, 4, 6: mean = 12/3 = 4 (changed), median = 4 (still the middle value — unchanged again), but now 2, 4, and 6 each appear exactly once — the original mode of 2 has lost its lead because the removed extreme took away one of its two occurrences.

So median is the one measure that is always unaffected by removing one extreme from each end of sorted data. Mean generally changes because it is recomputed from the shrunk sum and count. Mode may or may not change — it depends on whether one of its own occurrences happened to be exactly the extreme value being removed and, if so, whether that is enough to cost it the lead — so unlike median, neither mean nor mode carries a structural guarantee of staying the same.