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## Praxis Core Math

### Unit 1: Lesson 3

Statistics and probability- Data representations | Lesson
- Data representations | Worked example
- Center and spread | Lesson
- Center and spread | Worked example
- Random sampling | Lesson
- Random sampling | Worked example
- Scatterplots | Lesson
- Scatterplots | Worked example
- Interpreting linear models | Lesson
- Interpreting linear models | Worked example
- Correlation and Causation | Lesson
- Correlation and causation | Worked example
- Probability | Lesson
- Probability | Worked example

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# Center and spread | Worked example

Sal Khan works through a question on the center and spread of data from the Praxis Core Math test.

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## 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.