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Center and spread | Worked example

Video transcript

- [Instructor] So here we have a question on center and spread. It says the median and range of the mass of 12 panda cubs are 40 kilograms and seven kilograms, respectively. If both the largest and the smallest mass are removed from the data set, which of the following statements regarding the median and range of the 10 remaining masses could be true? It says choose all answers that apply. So pause this video, and see if you can have a go at this before we do it together. Okay, now let's do it together. So first, let's make sure we understand what they're saying. They're saying the median and range are 40 kilograms and seven kilograms, respectively, so that means whatever order they're saying median and range, that's the order of the data. So the median is 40 kilograms, so median is 40 kilograms, and then the range, range is seven kilograms. Now, before we go any further, let's remind ourselves what median and range even are. Median is, one way to think about it, is it's representative of the middle of a set of numbers. So let's say, I'm gonna do another example here before we look at the panda cubs. Let's say you have the numbers five, two, two, three and four. So the median here is, what you would do is you'd put them in order, so you'd go two, then we have another two, then you have your three, then you have your four, and then you have your five. So you put them in order and, if you have an odd number of numbers, the median is the middle number, just like that. And if you have an even number of numbers, it's halfway between the middle two numbers. That tends to be what people do. So, if you wanted to take the median of one, two, three, four, you have your middle two numbers are two and three. And then the median here would typically be 2.5, halfway in between two and three. Now what is the range? Well the range is the difference between your high and your low. So here, in this case, our range would be four minus one, so our range would be three. And here our range would be five minus two, so our range would be three again. So, what I would do to understand which of these could be true, let's think about these 12 masses, and let's come up with some numbers that meet these constraints and then think about what happens when you remove the high and the low, the largest and the smallest mass. So we have 12, so one, two, three, four, five, six, seven, eight, nine, 10, 11, 12. And since it's an even number of numbers, the median's gonna be by taking halfway between the middle two. So one, two, three, four, five, six. So the middle two are right over here. Halfway in between them, that needs to be 40 kilograms. Now the easiest way to do that, you can have the same mass multiple times, so let's just make that 40 and 40, just to meet the constraints. It could have been 39 and 41, or you could even use decimals, but this is an easy enough example. And then we want the range, we want the high minus the low to be seven kilograms, and then the high has to be at least 40 or higher, and the low has to be at most 40 or lower, and the difference between the two has to be seven kilograms. So let's make this one, let's say that this is 45 kilograms and then seven less than that would be 38 kilograms. So that meets all of the constraints that we're talking about. And then if we like, we can fill in these other numbers. So maybe we have another 38, maybe we have a 39 here, you have a 40 and then another 40. And remember we're putting them in order, so maybe we have another 40 here, maybe this is 41, 43, and then we have another 43. Now with this data set that we happened to construct that meets the constraints that they gave us, let's see what happens when we remove the high and the low. If we remove the low and the high, well that doesn't change the median. The middle two numbers are still going to be the same. And that's in general going to be true. As long as you remove both the high and the low, the median does not change, so does no change. Now what about the range? Well, before our range was seven kilograms. In this particular case, when we remove these two, now our new range is 43 minus 38, so our range has gone down. Our range has gone down to five kilograms. Now that doesn't always have to be the case. It definitely can be true that the range goes down, but you could have a situation where this was also a 45, in which case the range wouldn't change. But it's definitely possible for the range to go down. So what we see, and it's actually impossible for you to construct some numbers where the median changes at all. So the median, it's not possible that it could change. But it is possible that the range goes down. It's not possible for the range to increase because, in order for the range to increase, this number and this number would have to be further apart, but then all of a sudden that wouldn't be possible because this number is larger than this number, and this number is smaller than that number. So what's possible? The only thing that's possible is that the range could decrease or stay the same, and that the median would stay the same. All right, now let's look at the choices. The median and range will both decrease. Nope, the median cannot decrease. The median will stay the same, yup that can happen, but the range will decrease, yup that's possible. The median will stay the same, but the range will increase. Nope that's not possible. If they had another choice that said the median will stay the same and the range will stay the same, that would also be possible. And since this is a choose all answers that apply, that could have also been something that we circled. So big picture, we're talking about our measures of center, median is one of them, and we're also talking about our range, which is a measure of our spread. And in problems like this, when they talk about a data set, and they're not giving you all of the data, I recommend, come up with a data set that meets the constraints that they're giving, and then do what they're saying happens, and then see what happens to at least that particular data set, knowing that you just have a particular example, and so it's definitely valuable to figure out what could be true.