- [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.