Whether adding a value raises or lowers the median

0:00The median is the middle value. But middle of what? Not the middle of the range. Sort the numbers, and take the one standing in the middle of the line. Which works cleanly when there is an odd number of them, because then somebody really is in the middle. With an even number, nobody is. Two values share the middle, and the rule is that you average them - you do not pick one.

0:27So the count decides how you read it. And that is the only fussy part of this whole topic, because everything else follows from one small fact: the median only ever asks a value which SIDE it is on. Never how far away it is. Just which side. Five values: four, seven, eight, twelve, fourteen. Already in order. Five is odd, so the middle one is the third: eight. Count either side. Two values below eight, two above it. That balance of counts is exactly what being the middle means.

1:06Notice what the median is ignoring. The fourteen could be forty and eight would still be the middle value, because it would still have two on each side. Now a sixth value arrives, at eleven. Eleven is above eight, so recount. Below eight: still two. Above eight: now three. And that settles it. A value with more above it than below it is not in the middle of anything. Eight has lost the job.

1:38This is the whole argument, and notice that it never asked how big eleven was. Only that it landed on the upper side. So who is the middle now? Nobody - there are six values, and six is even. The two sharing the middle are eight and eleven. Average them: nine and a half. And nine and a half is not one of the six values. That is fine. The median is a position on the line, and with an even count it usually lands between two of your numbers rather than on one.

2:15The median has stepped up, from eight to nine and a half. Insert below instead and the same argument runs backwards. Put five in. Five is below eight, so now three values sit below eight and only two above it. Eight has too many below it, so eight is out. The two sharing the middle are seven and eight, and their average is seven and a half. Down, not up. And again the size did no work - a value at five and a value at minus five would have given exactly the same answer.

2:53There is a third case, and it is the easy one. What if the newcomer arrives sitting exactly on the median? Put another eight in. Now there are two eights in the middle, and the pair sharing the middle is eight and eight. Their average is eight. Nothing has moved. A value on the median is on neither side. It has no side to argue for, so it cannot pull the middle anywhere.

3:22Now the question that catches people out. How far can one new value move the median? Look at the neighbours of eight in the sorted line: seven below it, twelve above it. Add anything above the middle and the new median is the average of eight and twelve - or of eight and eleven, if the newcomer squeezes in between. Either way it lands between eight and twelve. Add anything below and it lands between seven and eight.

3:53So before you know a single thing about the newcomer, you already know the median will end up somewhere between seven and a half and ten. That is the entire range of possible answers. Try it. Eleven gives nine and a half. Twelve gives ten. A hundred gives ten. Ten thousand gives ten. The newcomer chooses the side. Your existing data chooses the size. Put that same hundred in and watch the other summary.

4:26The average of the five values was nine. Add a hundred and it becomes about twenty-four point two. The median went from eight to ten. The mean went from nine to nearly twenty-five. Same data, same insertion, two completely different reports - because the mean asked how far and the median asked which side. There is a word for the median's behaviour. Robust: unmoved by a value out at the end of the line.

4:57Which is a virtue when that far-out value is a mistake or a rarity, and a fault when it is the very thing you needed to notice. Now watch how far the side-counting idea can be pushed. Here are sixteen values, sorted. Sixteen is even, so the middle is shared by the eighth and the ninth - and both of them are forty-one. The median is forty-one, and the two middle values are the same number. That turns out to matter enormously.

5:31Add any single value you like. Seventeen values now, so the middle is the ninth. If your newcomer is small, it pushes everyone along and the ninth is a forty-one. If it is large, the ninth is still a forty-one. The median is forty-one. Whatever you added. And take any single value away instead - the same thing happens, for the same reason read backwards. So how do you move it? Two values, and not just any two.

6:04One below and one above, and the middle stays put - thirty and sixty, say. The two newcomers push from opposite ends and cancel. But send both from the same side and it shifts. Two values below drag it down to forty and a half. Two above push it up to forty-four and a half. The condition is not about their sizes. It is that one of them is at most forty-one and the other at least forty-one.

6:34Meanwhile every one of those four moves changes the average, because the average never had this defence in the first place. Turn it round one more time and let the median set the question. Ten values, and an eleventh you get to choose. You need the median to come out at twenty-nine. Eleven values means the middle is the sixth. Sort what you have, and twenty-nine is currently sitting sixth from the bottom of a list that is one short.

7:07So you need your value not to displace it. Anything at twenty-nine or above leaves twenty-nine in sixth place. Anything below pushes it to seventh, and something smaller becomes the middle. Given a list of candidates you could test them one at a time. Five of the seven work. But the condition is the better answer: your value must be at least twenty-nine. That is every solution there is, and it came from counting places, not from trying numbers.

7:42Two last things, and they are both about being careful. First: take a value away from BELOW the median. What happens? It might rise, and it might not move at all. It can never fall - because removing from below can only ever leave the middle with fewer beneath it. Never is a much stronger word than sometimes, and it is the true one here. Second, and harder. Twenty-four heights whose average you know, and two new people join at known heights.

8:14The new average is completely determined. The new median is not - and here are two sets of twenty-four with the very same average whose medians end up in different places. So the honest answer is that there is not enough information. That is not giving up. It is the correct answer, and recognising it is part of the skill. The median only ever asks which side. That is why it holds steady when one value runs away - and why, sometimes, it cannot be pinned down at all.