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Data representations | Worked example

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- [Instructor] We have a question here on data representation. It says which of the following dotplots represents a distribution in which the mean is less than the median? So pause this video and see if you can answer that. Okay, now before we do it together, let's just remind ourselves what the mean and the median are. The mean is what you normally associate in everyday language with the average. So if let's say I have four numbers, one, one, three, and four, if I wanna calculate the mean here, I add them all up, one plus one plus three plus four, and then I divide by the number of numbers there are. In everyday language, as I just mentioned, a lot of people would consider this the average. Now the median is the middle number. So if we take these same four numbers, if I were to figure out the middle number, I'd wanna order them. They're already ordered. One, one, three, four. And when you have an even number of numbers like this, you look at your middle two numbers, which in this case is one and three, and then what's typical is that you would go halfway between one and three. And so halfway between one and three would be two. So in this case, the median would be two. Now if you had an odd number of numbers, let's say you had one, one, one, three, and four, then you have a very clear middle number here. You order the numbers like this, and then the middle number is this one over here, so that would be the median. So with that out of the way, let's think about the mean and medians for these here. And we should be able to do it without a lot of computation. So let's look at choice A. So this tells us that we have three fours and four fives. And we could write it out if we like. So we have four, four, four. So we have those three fours. That's what this dotplot shows us. And then we have four fives. Five, five, five, five. And these are in order, so what is the middle number? The middle number is this five right over here. You have three numbers on either side of it. And so that is going to be the median. The median here is five, median is equal to five. Now what's the mean? Now you might be tempted to add up four plus four plus four plus five plus five plus five plus five and then divide by seven. That requires a lot of computation. But all you have to do to answer this question is to realize well look, you have some fours here, you have some fives, so the mean here is going to be between four and five. It's going to be a little bit closer to five since you have more fives, but it's going to be between four and five. So it's going to be less than the median. So this is a distribution in which the mean is less than the median. So I would choose this one right over here. Now let's look at this choice. In this choice, our median is going to be what? Well it goes the other way. You might be able to visually see it that your middle number is going to be a four, but we could write it out. We have four fours. Four, four, four, four. And we have three fives. Five, five, five. Our middle number, our median, is four. And our mean once again is going to be between four and five. In this case it's going to be a little bit closer to four. But this is a case where the mean is greater than the median. That's now what they're asking us. They're asking us the mean is less than the median. Rule that one out. This choice over here, what is going to be our median? Our median, let's see we have a three, and then we have four, then we have three fours, and then we have three fives. So then we have a five, a five, and a five. Our middle number is going to be four, and once again you might have been able to just see that visually. So your median is going to be four. Now what's your mean going to be? You might be tempted to calculate it like this, but you could say look, all right, the, you have these three fours here. Now you have a three that might bring the mean to the left, but then you have three fives that would more than bring that to the right. So this situation, because you have so many fives, you have more fives than you have threes, this would pull the mean to the right. So the mean once again is going to be some place around here. So your median is four, and your mean is greater than that four. So rule this one out. So this situation, your median, you might be able to pick it out right now. If you were to list all these numbers, your middle number is going to be four. So your median is four. Write that down, median is going to be four. What's the mean over here? Well, what you might realize is the mean of all of these fours is going to be four, and then the mean of this one three and this one five is going to be four again. Another way to think about it is that these balance each other out perfectly, and so the mean here is four. So this is a situation where the mean and the median are going to be the same, but that's not what they're asking for. And then we have another situation just like that, although it looks a little bit different. The middle number here is the four. And the mean is also going to be four. Because we have an equal number of numbers that are above four and below four, and they're just as far from four. We have three numbers that are one more than four, and we have three numbers that are one less than four. Or we have three data points that are one more than four at five, and we have three data points that are one less than four at three. So these are going to balance out, getting us back to a mean of four. So this is a situation again where the mean and the median are both four. Rule that one out, and we like choice A.