- [Charlie] In the previous screencasts
you saw how we can set up sampling distributions,
and that's gonna be really important
when we start to talk about hypothesis testing.
Another really important thing for understanding
hypothesis testing is the central limit theorem.
So, I'm gonna quickly explain
the central limit theorem in this screencast.
Let's say you have some process or calculation.
This could be a chemical reaction
or it could be some sort of mathematical modeling process,
and you've got all sorts of inputs, x one through xn,
and they affect x.
The output x is some function of those inputs.
We can have something that looks like this
where it's asymmetric.
We might have the uniform distribution,
or we might have something
that looks like an exponential distribution.
So, all of these distributions
that come in as inputs are not normally distributed.
The central limit theorem is really interesting
because as long as n,
the number of samples that you're taking in your output,
is sufficiently large, and that's in quotations.
I'll talk a little bit more
about what we consider large in a moment.
But as long as n is sufficiently big
the average x will be approximately normally distributed.
So, we can have inputs that are not normally distributed,
and as long as we're taking lots of samples in the output
to compute x bar, the average,
then the average is going to be normally distributed.
For example, here on the left
we might have an exponential distribution,
on the right we have the uniform distribution.
As we take larger samples from that, n equals two,
you start to see the distribution become more central.
N equals six is starting to appear
like the normal distribution.
And if we look at the sampling distribution
of x bar when n equals 25
it's very normally distributed in both cases.
So, the sampling distribution of both becomes normal
as the sampling size increases,
even though the inputs to this process
are clearly not normal.
What I showed in the previous screencast
on sampling distribution was that x bar
will be normally distributed as long as x one, x two,
all the way up to xn are each normally distributed,
and n can actually be quite small.
N could be three, for example, it can be pretty small.
And as long as all those inputs are normally distributed
x bar is also normally distributed
because linear functions of independent random variables
that are normally distributed are also normally distributed.
However, if we look at other inputs
that are not normally distributed,
so regardless of the distribution,
as long as the number of samples you're taking
is quite large the sampling distribution of x bar
will be approximately normally distributed.
And this is a unique property known
as the central limit theorem.
Thus, the probability distribution of x bar
can be determined using the standard normal distribution
where we can standardize using a Z-value, a Z-statistic.
We can calculate this as x bar minus mu
divided by sigma over root n.
This enables us to calculate the probability
of the average of population being x bar.
For example, what's the probability that the average
of 10 samples is less than 12, or greater than 20?
That's what this enables us to calculate.
Practical considerations for the use
of the central limit theorem,
whenever n is greater than or equal to 30
the normal approximation can always be used
even if we have clearly non-normal variables going into it,
like exponential or uniformly distributed variables.
For n less than 30 the normal approximation can be used
when distribution is not severely non-normal.
When comparing two independent populations, by the way,
the sampling distribution for the difference in sample means
can also be approximated by this formula.
So, if x one and x two are from different populations
then we can approximate the difference
in those two different means, the population means,
by using this formula where populations one and two
have different standard deviations or variances.
We can use this, the difference in means,
for various things.
For example, comparing the activity of two catalysts,
and later in the course we're gonna be doing
quite a bit on the difference in means.
You're going to be exploring the central limit theorem
in a workshop in Excel.
You're gonna essentially simulate
a bunch of random experiments,
and show that the output is normal
even if the inputs are non-normal.
Thanks for watching.