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## Mean and median challenge problems

## Video transcript

- [Voiceover] Let's think about what happens to the median and mean of a set of numbers when I change one of the numbers. And so let's look at this example. A group of four friends likes to bowl together, and each friend keeps track of his all-time highest score in a single game. Their high scores are all between 180 and 220, except for Adam, whose high score is 250. Adam then bowls a great game and has a new high score of 290. How will increasing Adam's high score affect the mean and median? Now, like always, pause this video and see if you can figure this out yourself. All right, so let's just think about what they're saying. We have four friends, and they each keep track of their all-time high score, so we're gonna have four data points, an all-time high score for each of the friends. So let's see, this is the lowest score of the friends, this is the second lowest, second to highest, and this is the highest scoring of the friends. So let's see. Their high scores are all between 100 and 220, except for Adam, whose high score is 250. So before Adam bowls this super-awesome game, the scores look something like this. The lowest score is 180. Adam scores 250. And if you take Adam out of the picture, the highest score is a 220. And we actually don't know what this score right over there is. Now, after Adam bowls a great new game and has a new high score of 290, what does the data set look like? Well, this guy's high score hasn't changed. This guy's high score hasn't changed. This guy's high score hasn't changed. But now Adam has a new high score. Instead of 250, it is now 290. So my question is, well, the first question is, does this change the median? Well, remember the median is the middle number. And if we're looking at four numbers here, the median is going to be the average of the middle two numbers. So we're gonna take the average of whatever this question mark is and 220. That's going to be the median. Now, over here, after Adam has scored a new high score, how do we calculate the median? Well, we still have four numbers, and the middle two are still the same two middle numbers. Whatever this friend's highest score was, it hasn't changed, and so we're gonna have the same median. It's gonna be 220 plus question mark, divided by two. It's gonna be halfway between question mark and 220. So our median won't change. So median, no change. Median, no change. So let's think about the mean now. Well, the mean, you take the sum of all these numbers and then you divide by four, and then you take the sum of all these numbers and divide by four. So which sum is going to be higher? Well, the first three numbers are the same, but in the second list you have a higher number. 290 is higher than 250. So if you take these four and divide by four, you're going to have a larger value than if you take these four and divide by four, because their sum is gonna be larger. And so the mean is going to go up. The mean will increase. The mean will increase. So median, no change, and mean, increase. All right, so this says both increase. No, that's not right. The median will increase. No, median doesn't change. The mean will increase. Yep. And the median will stay the same. Yep, that's exactly what we're talking about. And if you wanna make it a little bit more tangible, you can replace question mark with some number. You could replace it, maybe this question mark is 200. And if you try it out with 200, just to make things tangible, you're going to see that that is indeed going to be the case. The median would be halfway between these two numbers. And I just arbitrarily picked 200. It could be any number some place between 180 and 220, but you see, for this example, it's very tangibly that the median does not change. But the mean increases, 'cause we increase the sum. We increase the sum by 40, because we've increased this last number, and only the last number, by 40.