Statistics Learning Centre presents
Understanding the Normal Distribution.
Hi, I'm Dr Nic,
and in this video I'm going to explain
what the normal distribution is
and why it is so amazingly useful.
I will also give some examples of finding probabilities
in a normal distribution.
This video follows on from "Distributions 3:
All random variables have distributions.
When we think of probability distributions,
we tend to think of the normal distribution,
which is the most well known.
The normal distribution is really good
for modelling many natural processes,
manufacturing processes,
and human endeavours.
You have probably seen its bell shape many times.
The normal distribution has the following characteristics.
Single peak:
Most data values occur near the mean
so that there is a single peak in the middle.
Symmetry:
Half of the data values measured occur above the mean,
and half occur below the mean
so that the distribution is symmetric.
Tapers:
The further you get from the mean, the fewer people or things occur
so the distribution tapers out in both directions.
Caused by multiple factors:
Whenever there are many different factors affecting an outcome,
we tend to get something that is well modelled
by a normal distribution.
For example, if we look at the size of an apple,
that is affected by the sunshine, wind and rain on every day of its life.
Also, its genetics, pollination date, fertiliser,
and all sorts of other things.
So the distribution of the sizes of apples of the same variety
would probably resemble a normal distribution.
Each normal distribution is defined
by the mean and the standard deviation.
The mean is also the median
and tells us the midpoint of the distribution.
It is the value around which all the data values are centred.
The standard deviation tells us about how spread out the data is.
A useful rule of thumb is that most, or 95% of the data values,
will occur within 2 standard deviations of the mean.
It is important to note that there are also many things
that are not well modelled by the normal distribution.
Many distributions are not symmetric
and have considerable skew,
often to the right.
House prices, incomes, and human weights
are not well modelled with the normal distribution
as there can be quite a long tail to the right.
Even test scores in school are not always well modelled
by a normal distribution.
But, when you add together many occurrences
from even non-normal distributions,
you tend to get a normal distribution pretty quickly.
Here is an example of what I mean.
This is a distribution of number of ice creams purchased per customer
at Luke's ice cream stand.
It is based on data he has collected over the last few weeks.
It is clearly nothing like a normal distribution.
We run a simulation of 1000 customers
and see what the distribution looks like.
It is similar to the original,
as we would expect.
What say we want to look at what 1000 lots of 2 customers buy,
or 1000 lots of 3 customers,
or 4 customers, or 10.
We see that the shape of the graph starts to look more and more
like a normal distribution.
For 10 customers to order a total of 30 or more ice creams,
they would all need to order about 3 ice creams each,
which is pretty unlikely.
Similarly, it is pretty unlikely that all 10 people will order
just 1 ice cream for a total of 10.
Most of the time, the values will be much less extreme.
So, even though the distribution for each individual customer
is nothing like the normal,
once we add even 10 customers together,
the resultant distribution closely resembles the normal distribution.
That is why the normal distribution is so useful.
We have looked at the bell-shaped graph a lot in this video,
and now I'm going to explain how it works.
That bell-shaped graph is called the PDF, or probability density function.
To find the probability that an outcome is within a certain range,
we look at the area under the graph within that range.
We could think of it as the probability that a randomly chosen pixel
occurs within that range.
Here is the PDF of a normal distribution
approximating the weight of Luke's ice creams.
There is a mean of 112 grams,
and a standard deviation of 9 grams.
Straight off, we can say that 95% of all of Luke's ice creams
will lie within 2 standard deviations of the mean;
this is always true for normal distributions.
So we say that 95%, or most, ice creams
weigh between 112 - 18 which is 94 grams,
and 112 + 18 which is 130 grams.
You can see that 95% of the area under the graph
is between 94 grams and 130 grams.
We can find out the probabilities of other values too.
Luke wants to know what proportion of his ice creams
are likely to weigh less than 100 grams.
So we want to find the probability that X is less than 100,
which is represented by the area under the graph
to the left of the line going up from the number 100.
We can use a calculator, a spreadsheet or tables,
and find the value 0.091.
You can see that .091, or 9% of the area,
is to the left of the number 100.
We would expect 9% of ice creams to weigh less than 100 grams.
You might want to know how likely it is
that an ice cream weighs exactly 120 grams.
Strictly speaking, the answer is zero
as the probability of any exact value is zero
in a continuous distribution.
This is because there are infinitely many ice cream weights
between, say, 119.9 and 120.1.
So the probability of any specific value
is 1/∞, which is zero.
For example, an ice cream could weigh 120 grams,
or 120.000001 grams.
But, our scales are only accurate to the nearest gram,
so they will read 120 grams for any ice cream weighing
between 119.5 grams and 120.5 grams.
We can find the probability of an ice cream weight's
being in that interval.
And the answer to that is 0.03.
Practically speaking, 3% of the ice creams
will be measured as weighing 120 grams.
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