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# Center, spread, and shape of distributions — Basic example

## Video transcript

mr. Jadhav raised all of his students scores on a recent exam by 10 points what effect did this have on the mean and median of the scores now there's two ways we could approach this one is we could come up with a simple example that meets the constraints a plausible combination of scores and then see what what is going to be true which of these statements get contradicted or don't get contradicted that's one technique and that might actually be a simpler one if you're under time pressure under in something like the SAT and another technique is to do it a little bit more rigorously so let's do the simple way first so when you say okay you know his students you imagine a classroom with twenty or thirty students but they don't say it's twenty or thirty students you could imagine that it could be three students and if it could be three students and these statements need to be true for any of the plausible combinations of scores so let's just think a simple one let's just imagine that the three students all got 80s and I picked I just randomly picked those numbers because it's very easy to calculate both the median and the mean here the mean here is 80 and the middle the middle score here is the median is also 80 so both median median is equal to 80 here and the mean is equal to 80 now if you add a 10 to all of these if you add a 10 to all of these then it becomes a 90 a 90 and a tiny and then your median and mean are both going to become 90 median is going to be equal to the mean which is going to be equal to 90 so at least for this case which is plausible it's not necessarily the case that they're talking about but for this case when you increased by 10 but when you increased all the scores by 10 both the median and the mean increased by 10 now let's see what these statements say the mean increased by 10 points but the median remained the same well this combination we just said this was plausible this is this is this could have been mr. Jat of students test scores but it contradicts the statement the median didn't remain the same in this case right over here so we could cross that out the median increased by 10 points but the mean remain the same well once again this little case that we came up with it cut its applause well that was the scores of his students but it contradicts this statement so the statement is definitely not true for all the possible combinations of his students test scores the mean increased by 10 points and the median increased by 10 points now this combination we picked this this particular case we picked this doesn't prove that this is always going to be true but at least it doesn't contradict it so we can't cross it out just yet the mean and the median remain the same well we were able to come up with this case which is plausible this could have been his student scores and it contradicts this in order for us to be able to select a statement we have to feel good that it would be true for any combination of scores that his students had so we could cross that out as well so if I'm under time pressure I'm taking the SAT I would definitely do this choice and then move on now I'm sure a lot of you probably want a little bit more of a rigorous proof that we could say like for any combination of scores the mean would increase by 10 points and the median would increase by 10 points and for that we could do a little bit of a of a justification borderline proof right over here so let's just imagine you know this is the score 1 and then this right over here is the median score and then we keep going and then this is the enth that is the nth score right over there now if we added if we added 10 to everything the order isn't going to change so all of these scores these end scores are just going to go up by 10 so this is going to be s 1 plus 10 and then you have s 2 plus 10 and then you're gonna have your former median plus 10 but that's still going to but now this thing is going to be the middle value and then you're gonna have your highest value I'm assuming that I ordered these from lowest to highest and then you're going to have your high score plus 10 so if all the scores go up by 10 whatever was the the kind of that median that's still gonna be in the middle but now it's gonna have be 10 higher so your new median is going to be 10 higher so hopefully this justification shows you that for whatever combination of scores if you order them in this way and you need to order them to figure out the middle value the median value that the median indeed would go up by 10 now let's feel good that the mean would go up by 10 so how would you calculate the mean so the mean is going to be the first score plus the second score all the way to the nth score and you're going to divide it by n that is what that is going to be equal to the mean now if you added 10 to all of these if it was s1 plus 10 and plus s2 plus 10 and we went all the way to SN plus 10 divided by n well what's that gonna be what's that gonna be well we take all the tens out we have we're adding ten n times so we could rewrite this as s 1 plus s 2 plus s n and then we have ten plus ten n times so we could write plus 10 n and then divide that whole thing divide that whole thing by n I just rewrote this I just instead of right adding the 10 n times I just wrote 10 times N or n times 10 well this right over here I could instead write this like this this is going to be that divided by n plus that divided by N and what is this value right over here this is your mean your your old mean + 10 n divided by n well it's gonna be plus 10 so your new mean your new mean is going to be the old mean plus 10 so hopefully that gives you a justification not just using a special case to rule out that contradicts these other cases but a justification why this would be true for all combinations of scores we could feel very good about this