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AP.STATS:

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CCSS.Math: , - [Instructor] One of the
most commonly used tools in all of statistics is
the notion of a Z-score. And one way to think about a Z-score is it's just the number
of standard deviations away from the mean that
a certain data point is. So let me write that down. Number of standard deviations. I'll write it like this. Number of standard deviations
from our population mean for a particular, particular data point. Now let's make that a little bit concrete. Let's say that you're some
type of marine biologist and you've discovered a new
species of winged turtles and there's a total of
seven winged turtles, the entire population of
these winged turtles is seven. And so you go and you're actually able to measure all the winged turtles and you care about their
length and you also wanna care about, how are
those lengths distributed? Lengths of winged turtles. All right, and let's say, and
this is all in centimeters. These are very small turtles. So you discover, and these are all adults. So there's a two centimeter one, there's another two centimeter one. There's a three centimeter one. There's another two centimeter one. There's a five centimeter
one, a one centimeter one, and a six centimeter one. So we have seven data
points and from this, and I encourage you at
any point if you want. Pause this video and see
if you wanna calculate, what is the population mean here? We're assuming that this is the population of all the winged turtles. Well, the mean in this situation
is going to be equal to, you could add up all these
numbers and divide by seven and you would then get three. And then using these
data points and the mean you can calculate the
population standard deviation. And once again, as review
I always encourage you to pause this video and see
if you can do it on your own. But I've calculated that ahead of time. The population standard
deviation in this situation is approximately, I'll round
to the hundredth place, 1.69. So with this information you
should be able to calculate the Z-score for each of these data points. Pause this video and
see if you can do that. So let me make a new column here. So here I'm gonna put our Z-score. And if you just look at the definition what you're going to do for
each of these data points, let's say each data point is x, you're going to subtract
from that the mean and then you're going to divide that by the standard deviation. The numerator right over
here's gonna tell you how far you are above or below the mean, but you wanna know how
many standard deviations you are from the mean,
so then you'll divide by the population standard deviation. So for example, this first
data point right over here if I wanna calculate the
Z-score I will take two. From that I will subtract three and then I will divide by 1.69. I will divide by 1.69. And if you've got a calculator out this is going to be -1 divided by 1.69 and if you use a calculator you would get, this is going to be approximately -0.59. And the Z-score for this data
point is going to be the same. That is also going to be -0.59. One way to interpret this is, this is a little bit more
than half a standard deviation below the mean, and we could
do a similar calculation for data points that are above the mean. Let's say this data point right over here. What is its Z-score? Pause this video and see
if you can figure that out. Well, it's going to be six
minus our mean, so minus three. All of that over the standard deviation. All of that over 1.69 and
this, if you have a calculator, and I calculated it ahead of time, this is going to be approximately 1.77. So more than one, but less
than two standard deviations above the mean. I encourage you to pause this video and now try to figure out the Z-scores for these other data points. Now, an obvious question that
some of you might be asking is why, why do we care how
many standard deviations above or below the mean a data point is? In your future statistical life, Z-scores are gonna be a really useful way to think about how usual or how unusual a certain data point is. And that's going to be really valuable once we start making
inferences based on our data. So I will leave you there. Just keep in mind it's a very useful idea, but at the heart of it
a fairly simple one. If you know the mean you
know the standard deviation. Take your data point, subtract
the mean from the data point and then divide by your
standard deviation. That gives you your Z-score.

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