This video leads to introducing the normal distribution.
But to get to that point, which we will in the next video,
we first have to consider the Central Limit Theorem (CLT),
as well as the statistical idea of independence.
The Central Limit Theorem (CLT) is an important theorem in Statistics.
It is stated as follows: take a sample of values from any distribution and average them.
That average value will be as if it was from the normal distribution.
There is a restriction that the sampled data come from a distribution of finite variance.
There are a lot of words and new concepts there, so let's break it down a bit.
We are going to take N samples and calculate the average; we call that average x-bar.
The CLT says x-bar comes from the normal distribution.
This theorem is so relevant in science and engineering work,
as well as areas where data analysis is used.
That's because the concept of using averages appears in most data analysis.
The raw data used in those samples comes from unknown distributions.
If you were to go to your process and measure the thickness of boards produced,
those boards could come from a distribution that we don't know
about (it might be an F-distribution, or a uniform distribution).
That average board thickness value however, is normally distributed though:
we don't even need to confirm it.
It's guaranteed for us by this theorem.
That's useful, because as we will see in future videos, we can answer all sorts
of interesting questions about data that comes from a normal distribution.
There are two restrictions though: the first one is satisfied for all practical systems,
and that's the requirement the data come from a distribution of finite variance.
The one that is more concerning is the requirement that the samples are independent.
We are more likely to violate that last restriction.
Let's explain the CLT visually now, using the idea of throwing dice.
Let's use a regular dice with 6 numbers.
Each throw is a sample, and each throw is also independent of the other.
That means, the value I read on the dice on the current throw does not depend on the values
from prior throws, nor does it influence values I will get in the future.
Here is the histogram then of 500 throws, and because 500 divided by 6 is about 83,
we see bars which are all around that height.
We call this sort of distribution a uniform distribution.
Uniform distributions have finite variance.
So we meet all the restrictions of the CLT.
Now let's throw the dice twice and calculate the average of both numbers.
Plotting the histogram of those averages I get this distribution in the top middle row.
Throwing 4 times and then averaging those 4 numbers leads
to this distribution in the top right.
I could do this for averaging over 6, 8 or 10 throws and each
of these histograms are shown here in the bottom row.
These distributions are more and more normally distributed.
I'll jump ahead a little, and show you the classic bell-curve shape
of the normal distribution.
You've seen that before, and our histograms certainly appear to be similar.
It doesn't take very many points that you have to add up and average before you get averages
that start to appear normally distributed.
Before we go and look at the details of the normal distribution though I want
to look at the concept of independence.
It is an important assumption, used in other statistical work as well.
When we say we are sampling independently,
it implies there is no possible relationship between any two samples taken.
The current sample won't influence the value of a sample taken later on;
and the current sample is also not affected by any other prior sample we have already taken.
Let's look at some examples.
If I asked you to measure, in millimetres, the amount of snow on the ground, every 6 hours.
Will those values you take over period of say, about 1 week, be independent values?
If I asked you to get the amount of snow on the ground, on 15th January every year, since 1976,
will those values be independent?
In the former case the snow amounts are going to be related to each other,
since snow on the ground persists for a long time.
However snow values that are a year apart will have almost no relationship
to each other, given the separation in time.
In many engineering processes the samples might well have a relationship between each other,
because the speed at which our processes and systems move is often very slow.
Please bare that in mind in your career when you are analyzing data from those systems.
Here's another example: if I give a survey to you all in the class,
then I will get independent responses, or measurements, from you all.
But if you discuss that survey in groups, before submitting your responses,
I am not going to get independent measurements from you all.
Here's another case: consider these 3 sequences of data.
Which of these 3 plots show independent samples?
When I ask students about this, in all my prior classes, the majority of people state
that sample 3 shows an independent sequence.
The reasoning given is that it "just looks that way".
People reject sample 1 as being independent, because you can see trends in the data:
it moves slowly up and down, and therefore violates the requirement
that samples have no relationship with each other.
Take a look though a little more closely at sample 3, and you should observe the values seem
to oscillate: one sample is low, then the next is high, then back to low again; and so forth.
We have been fooled by our eyes here.
This sample is not independent either.
The key message here is that you cannot judge independence just by looking at the data.
We have to test for it, or we have to structure our sampling approach in a way
that we ensure we get independent values.
By the way, sample 2 was the one that was independent.
So let's recap here.
The average of a sequence of numbers will be normally distributed,
provided we meet two conditions: the samples are independent, and the samples come
from a distribution of finite variance.
In the next videos we will use the normal distribution, learn more about it's properties,
and how to answer interesting questions about normally distributed data.