Statistics Learning Centre presents: Understanding the central limit theorem.
Hi, I'm Dr Nic and I'm going to talk about the central limit theorem
including four important aspects.
This video complements our video,
"Calculating a Confidence Interval for a Mean using a formula."
It would also be useful to watch our video
Understanding the normal distribution.
The central limit theorem was first suggested several centuries ago
and many traditional statistical tests and procedures are based on it.
The central limit theorem is concerned with
the sampling distribution of the mean.
The sampling distribution is the
distribution of means of samples taken from a population.
Usually we do not know
the mean of the population so we take one sample
We use what we know or find out from that sample
to make an inference about the mean of the population.
We use the mean of the sample, x-bar,
the size of the sample n and the standard deviation of the sample, s.
I'm going to talk about four aspects of the central limit theorem.
Aspect one: the sampling distribution of the mean will
be less spread than the values in the population from which the sample is drawn.
Aspect 2: the sampling distribution will be well modeled by a
normal distribution
Aspect 3: the spread of the sampling distribution is
related to the spread of the population values
Aspect 4: bigger samples lead to
a smaller spread in the sampling distribution
I will now illustrate these aspects with an example.
We have a population of dragons
and we wish to know their mean strength in order to tell whether the
change in climate is affecting them.
We want to compare that with information
from previous scientific studies of the dragons.
Each dragon is represented by one of these data cards
From experience we know that the range of dragon strengths goes from 1 to 8.
There are 720 dragons in the population
and we're taking a sample of four dragons.
These are very small samples but are useful for illustrative purposes.
How likely is it that the average strength
of the four dragons will be one?
You are right! It is very unlikely.
All four dragons would need to have a strength of one
for the mean strength to be one.
If there are equal numbers of dragons at each strength then a probability of
getting four dragons of strength one is one-eighth to the power of four which
equals 0.00024 or one chance in 4096.
Similarly it is very unlikely that the mean strength will be eight
as all four dragons would need to have a strength of eight
Let us take some samples of our dragons and see what
the mean strength is for each sample.
Here's our first sample: the mean is 4.25
and here is another sample. The mean is 5.25.
and here's another sample. The mean is 4.75
and here are a few more samples
The means are 6, 5, 3.5 and 4.5
It could take a long time doing this by hand so let's set up our computer to
simulate taking 4,000 samples of four dragons from our population.
We can only do this because we happen to know all about the dragons strengths
having made them up in the first place.
This is a simulation.
We would never know this in a real-life study.
Here is the distribution of those four thousand means from
samples of four dragons.
Let's look at the four aspects mentioned before.
Aspect 1: the sampling distribution will be less spread than the population from
which it is drawn.
You can see that most of the sample means lie between three
and six and all of them lie between 1.75 and 7.5
Remember the population values lie between one and eight.
Aspect 2: the sampling distribution will be well-modeled by a normal distribution.
You can see that even with samples of size four
that the distribution resembles a normal distribution with a bell shape
If we take larger samples the distribution gets closer
in shape to a normal distribution
Here are graphs showing the spread of sample means
of dragons from samples of size 16 and 50
you can see that the shape is getting more like the normal distribution.
This is also shown in our video about the normal distribution
Aspect 3: the spread of the sampling distribution
is related to the spread of the values in the population
Say a different dragon population had a spread of strengths of
1 to 20 instead of 1 to 8
You would expect the spread of the sample means to be greater.
Here is a graph showing the spread of sample means from samples of
size 4 from a different population of dragons.
These dragons have strengths evenly spread between 1 and 20.
You can see that there is still the bell shape
but for these dragons most of the sample means lie between 6 and 16.
Aspect 4: bigger samples lead to a smaller spread in the sampling distribution.
The bigger the sample
the less likely it is that all of the values will be small
or that all of the values will be large
so that the chance of extreme values is reduced
Here is a graph showing the spread of sample means of strengths of dragons
from samples of size 25,
and here is a graph showing the spread of sample means of strengths of dragons from samples of size 50
You can see that the spread reduces as the sample size increases.
The spread of the sampling distribution is
related to the square root of the sample size.
Two important notes
You only ever take one sample from a population
We took thousands of samples to illustrate
what a sampling distribution looks like.
The central limit theorem only works under some conditions
and with big enough sample sizes.
Four is actually far too small
though you could still see the effect.
Under most conditions samples of 30 are sufficient.
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