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Median & range puzzlers

Sal solves an interesting median and range challenge problem.

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Video transcript

- [Voiceover] Let's say that we run a car dealership, and we have 11 salespeople, and we record the number of cars each of the 11 salespeople sold in the past week. So that's one of them, two, three, four, five, six, seven, eight, nine, 10, 11. So we record how much each of them sold. So maybe one person sold, maybe they sold five cars. Maybe the next person sold seven cars. Maybe the next person sold 10 cars. And so that's how we would do it. So we're going to record all of that. But then we're told that the median car sold is six. So if we ordered all of these numbers, so let's just assume that we ordered them all from least to greatest. So maybe, and I'm just making up numbers here, maybe this is four. This is four. Maybe this is five, five, five, six, seven, seven, eight, nine, 10. The median car sold, meaning that the middle number here is going to be six. Now we don't know whether all of these other numbers are the actual numbers. I just made those up. But we know that the middle number needs to be six. So let me just fill out only the middle number. So the middle number, if you have 11 data points, the middle number's going to have five on either side. So that's that one right over there. One, two, three, four, five. One, two, three, four, five. So we know that the median, we know the median-- Actually, let me do that a purple color. The median is going to be six. Now the other thing we know is that the range of cars sold is four. Now let's remind ourselves, the range is the maximum number of cars sold minus the minimum number of cars sold, and if we have sorted all of these out, if we take-- If this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be our max minus our min. So let me write that down. So you take your min and your max. You have your range. Range is equal to the maximum minus the minimum. And they're telling us that that is four. This range is four. So the maximum minus the minimum, the difference between the number of cars that the most productive salesperson sold and the number of cars that the least productive salesperson sold, that difference is going to be four. So given all of those assumptions, giving all of that information, I'm now going to give you a statement, and our challenge is if we assume everything I just said is true, is this statement going to be true, or is this statement going to be false, or do we not know? So let me write this down. So is this going to be true, false, or do we not know? Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? And so I encourage you to pause the video now and try it out. All right, so let's work through it together. This is kind of a fun puzzle here, where we've been given some clues, and then we have a statement, and we figure out can we make this statement. Can we say it's true or can we say for sure that it's false, or do we just not know? So a couple of ways to tackle it. One way to tackle it is, well look, one of the salespeople must have sold six cars. If the median sold is six, and there's 11 salespeople here, the middle number here, that if we order it, one of them must have sold six cars. If we had an even number here, then that would be the average of the two middles, but if we have an odd number here, that is literally the middle number of cars sold. So someone sold six cars. So if we assume-- We're trying to find a world where someone sold more than 10 cars. So if someone sold six cars, is there a way, with a range of four, that someone sold more than 10? Well let's just think about it. Let's just assume that the min is six. We don't know that the min is six, but let's just try it out. If the minimum is six, if the minimum is six, what's the maximum going to be? Well remember, the range is equal to-- We know that the range is four. The range is equal to the maximum minus the minimum, maximum minus the minimum, and so the maximum in this case would be 10, would be 10. The max would be equal to, would be equal to 10. If the min is equal to six, and instead of colons I'm going to write an equals sign, if the min is equal to six, then the max at most can be equal to 10. And we can't take a higher min, because we know that the six is going to be one of the values. We could try a lower minimum value. We could say the minimum value is five or four or three or two or one, but then the maximum would go down even more, because remember, the maximum is no more than four larger than the minimum. So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that any of the salespeople could have sold is 10. And so the statement, at least one of the salespeople sold more than 10 cars, that's got to be false. That's got to be false. Now there's another way that you could think about it. You could assume that someone sold more than 10. You could assume. In the last example, or the last way of thinking about it, we assumed that the min was six, but now let's just assume, let's just assume that someone sold-- Let's just try it out. Let's see if it's possible. Let's just assume that the max is 11, that someone sold more than 10 cars. If the max is equal to 11, what's the min going to need to be? Well we just have to remind ourselves, range is equal to max minus min. So four is equal to the maximum, 11, minus the minimum. So 11 minus what is equal to four? Well 11 minus seven is equal to four. So if we assume that the maximum is 11, then the minimum is going to have to be, the minimum is going to have to be equal to seven. Now can the minimum be seven when your median is six? No. If your median is six, that means you have-- That means if you have an odd number of data points, that means one of the data points is six, and if you had an even number of data points, that means that the middle two are going to average out to be six, which means you even have data points that are less than six. So your minimum can't be seven. You're going to have a value that is at least as low, at least as low as six. So your assumption can't be true, can't be true, so once again, the assumption based on the statement can't be true. This statement is going to be false.