Let's say that you're curious about people's TV
watching habits.
And in particular, how much TV do people in the country watch?
So what you are concerned with, if we imagine
the entire country-- and we've already talked about--
especially if we're talking about a country like the United
States, but pretty much any country,
is a very large population.
In the United States, we're talking
about on the order of 300 million people.
So ideally, if you could somehow magically do it,
you would survey or somehow observe all 300 million people
and take the mean of how many hours of TV
they watch on a given day.
And then that will give you the parameter, the population mean.
But we've already talked about, in a case
like this, that's a very impractical.
Even if you tried to do it, by the time you did it,
your data might be stale because some people might have passed
away, other people might have been born.
Who knows what might have happened.
And so this is a truth that is out there.
There is a theoretical population mean
for the amount of the average or the mean hours
of TV watched per day by Americans.
There is a truth here at any given point in time.
It's just pretty much impossible to come up
with the exact answer, to come up with this exact truth.
But you don't give up.
You say, well, maybe I don't have to survey all 300 million
or observe all 300 million.
Instead, I'm just going to observe a sample,
right over here.
And let's say, to make the computation simple,
you do a sample of six.
And we'll talk about later why six might not
be as large of a sample as you would like.
But you survey how much TV these folks watch.
And you find one person who watched 1 and 1/2 hours.
Another person watched 2 and 1/2 hours.
Another person watched 4 hours.
And then you get one person who watched 2 hours.
And you get two people who watched 1 hour each.
So given this data from your sample,
what do you get as your sample mean?
Well, the sample mean, which we would denote by lowercase
x with a bar over it, is just the sum
of all of these divided by the number of data points we have.
So let's see we have 1.5 plus 2.5 plus 4 plus 2
plus 1 plus 1.
And all of that divided by 6, which
gives-- let's see, the numerator 1.5 plus 2.5 is 4, plus 4 is 8,
plus 2 is 10, plus 2 more is 12.
So it's going to be 12 over 6, which
is equal to 2 hours of television.
So at least for your sample, you say,
my sample mean is two hours of television.
It's an estimate.
It's a statistic that is trying to estimate
this parameter, this thing that's very hard to know.
But it's our best shot.
Maybe we get a better answer if we get more data points.
But this is we have so far.
Now the next question you ask yourself is,
well, I don't want to just estimate my population mean.
I also want to estimate another parameter.
I also am interested in estimating my population
variance.
So once again, since we can't survey
every one in the population, this
is pretty much impossible to know.
But we're going to attempt to estimate of this parameter.
We attempted to estimate the mean.
Now we will also attempt to estimate
this parameter, this variance parameter.
So how would you do it?
Well, reasonable logic would say, well, we maybe we'll
do the same thing with a sample as we
would have done with the population.
When you're doing the population variance,
you would take each data point in the population,
find the distance between that and the normal population mean,
take the square of that difference,
and then add up all the squares of those differences,
and then divide by the number of data points you have.
So let's try that over here.
So let's try to find-- take each of these data points,
and find the difference-- let me do that
in a different color-- each of these data points,
and find the difference between that data point
and our sample mean-- not the population mean,
we don't know what the population mean--
the sample mean.
So that's that first data point plus the second data
point-- so it's 4 minus 2 squared plus 1 minus 2 squared.
And this is what you would have done
if you were taking a population variance.
If this was your entire population,
this is how you would you find a population mean here,
if this was your entire population.
And you find the squared distances
from each of those data points and then divide
by the number of data points.
So let's just think about this a little bit.
1 minus 2 squared.
Then you have 2.5 minus 2-- 2 being
the sample mean-- squared.
Let me see, this green color.
Plus 2 minus 2 squared.
Plus 1 minus 2 squared.
And then maybe you would divide by the number of data points
that you have, where you have the number of data points.
So in this case, we're dividing by 6.
And what would we get in this circumstance?
Well, if we just do the computation, 1.5 minus 2
is negative 0.5.
We square that.
This becomes a positive 0.25.
4 minus 2 squared is going to be 2 squared, which is 4.
1 minus 2 squared-- well, that's negative 1
squared, which is just 1.
2.5 minus 2 is 0.5 squared, is 0.25.
2 minus 2 squared-- well, that's just 0.
And then 1 minus 2 squared is 1, it's negative 1 squared.
So we just get 1.
And if we add all of this up-- let
me add the whole numbers first.
4 plus 1 is 5, plus 1 is 6, and then we have two 0.25s.
So this is going to be equal to 6.5--
let me write this in a neutral color.
So this is going to be 6.5 over this 6 right over here.
Well, there's a couple of ways we could write this,
but I'll just get the calculator out
and we can just calculate it.
So 6.5 divided by 6 gets us-- if we round,
it's approximately 1.08.
So it's approximately 1.08 is this calculation.
Now what we have to think about is
whether this is the best calculation, whether this
is the best estimate for the population variance, given
the data that we have.
You can always argue that we could have more data.
But given the data we have, is this the best calculation
that we can make to estimate the population variance?
And I'll have you think about that for a second.
Well, it turns out that this is close,
this is close to the best calculation, the best estimate
that we can make, given the data we have.
And sometimes this will be called the sample variance.
But it's a particular type of sample variance
where we just divide by the number of data points we have.
And so people will write just an n over here.
So this is one way to define a sample variance in an attempt
to estimate our population variance.
But it turns out-- and in the next video
I'll give you an intuitive explanation
of why it turns out this way.
And then I would also like to write a computer simulation
that, at least experimentally, makes
you feel a little bit better.
But it turns out, you're going to get a better estimate--
and it's a little bit weird and voodooish at first
when you first think about it-- you're
going to get a better estimate for your population variance
if you don't divide by 6, if you don't divide
by the number of data points you have
but you divide by one less than the number of data
points you have.
So how would we do that?
And we can denote that as sample variance.
So when most people talk about the sample variance,
they're talking about the sample variance
where you do this calculation, but instead of dividing by 6
you were to divide by 5.
You would divide by 5.
So they would say you divide by n minus 1.
So what would we get in those circumstances?
Well, the top part is going to be the exact same thing.
We're going to get 6.5.
But then our denominator, our n is 6.
We have 6 data points.
But we're going to divide by 1 less than 6.
We're going to divide by 5.
And 6.5 divided by 5 is equal to 1.3.
So when we calculate our sample variance with this technique,
which is the more mainstream technique--
and it seems voodoo.
Why are we dividing by n minus 1,
wherein for a population variance we divide by n?
But remember we're trying to estimate the population
variance.
And it turns out that this is a better estimate.
Because this calculation is underestimating
what the population variance is, this is a better estimate.
We don't know for sure what it is.
These both could be way off.
It could be just by chance what we happen to sample.
But over many samples-- and there's many ways
to think about it-- this is going
to be a better calculation.
It's going to give you a better estimate.
And so how would we write this down?
How would we write this down with mathematical notation?
Well, remember, we're taking the sum.
And we're taking each of the data points.
So we'll start with the first data point
all the way to the nth data point.
This lowercase n says that, hey, we're looking at the sample.
If I have an uppercase N, that usually
denotes that we're trying to sum up
everything in the population.
Here we're looking at a sample of size, lower case n.
And we're taking each data point, so each x sub i,
and from it we're subtracting the sample mean.
And then we're squaring it.
We're taking the sum of the squared distances.
And then we're dividing, not by the number of data points
we have, but by 1 less than the number of data points we have.
So this calculation, where we just summed up all of this
and then we divided by 5, not by 6,
this is the standard definition of sample variance.
So I'll leave you there.
In the next video, I will attempt
to give you an intuition of why we're dividing by n minus 1
instead of dividing by n.