Now that we understand how to calculate, mean variance and standard deviation for a normal data set,
we want to talk about the equation of the probability density function of the normal distribution.
So this function F of x is the standard form of the probability density function for a normal distribution.
Notice that it relies upon the standard deviation given here by Sigma.
We also see it here and the mean MU, which we see here.
So in order to build the equation for the probability density function, we need to have the mean and
standard deviation that we learned to calculate earlier.
So let's say, for instance, that we had from our dataset calculated a mean of five and a standard
deviation of three, then we could plug these values into the probability density function and we would
get one over three times the square root of two pi.
And then Euler's number here raised to the negative one half power multiplied by X minus our mean of
five, so x minus five, all divided by standard deviation, so three quantity squared.
And now we have here the equation for the probability density function of the normal distribution that
has a mean of five and a standard deviation of three.
Now we talked about earlier how a normal distribution was a bell shaped curve.
Well, here are some examples of these probability density functions we've been talking about.
In fact, this distribution in green here is the distribution for the normally distributed data with
mean five and standard deviation three.
In red, we have a mean of five and a standard deviation of two.
And in blue we have a mean of five and a standard deviation of one.
We can see that the mean is five for all three distributions because we see here that all three distributions
are centered at five.
Remember that for any normal distribution, the mean is right here at the center of the curve and for
any normal distribution, this is also where we'll find the median.
In other words, in a normal distribution, the mean and median are always equivalent.
Now what we notice is that the curve, the graph of the probability density function is of course just
dictated by these two values, the mean and the standard deviation, because those are the only two
values we need in order to plug into this formula for the probability density function.
And that's why we can express any normal distribution as just its mean and standard deviation, or what
we actually use for our standard notation, which is variance of course, just the square of standard
deviation.
So whenever we're dealing with any normal distribution, we will often just express it this way the
normal distribution of the mean comma, the variance.
And so instead of having to express this entire probability density function or give extra information
or a sketch of the curve, all we have to do is say we're dealing with the normal distribution.
Five one and that indicates a mean of five and a variance of one.
Here we have a normal distribution with a mean of five and a variance of four, and here we have a normal
distribution with a mean of five and a variance of nine.
Now we always give the variance, but of course we know that standard deviation is just the square root
of variance.
So based on these expressions of these three normal curves which correspond to these three curves that
we already sketched, we know that the standard deviation here is one, two and three, the square root
of one, four and nine respectively.
So just from this notation here, we can immediately see mean variance.
We can take the square root of variance to get standard deviation, and then we can use this information
to plug into the equation of the probability density function for a normal curve and use that equation
then to sketch the curve.
If we graph this equation after we've plugged in the mean and the standard deviation, we will get this
picture of the normal curve.
Let's notice again here too, that as the variance and standard deviation increase, the normal curve
gets flatter and wider.
And that's because variance and standard deviation are measures of spread.
And so when we see a higher value for variance and therefore a higher value for standard deviation,
that means the data is spread out further from the mean.
The data gets pushed out further away from the mean to both the right and left, as opposed to when
variance and standard deviation are smaller.
And we see more of the area under the curve, more of the data pushed closer to the mean on the right
and the left hand side.
That data is pulled in toward the mean.
And so we see the curve rise up and become taller.
That normal curve is taller.
The data is more tightly clustered around the mean.
The spread is smaller, but as long as our data.
Creates this symmetric bell shape following this form of a probability density function.
We know that we have normally distributed data regardless of whether the curve is taller and more narrow,
like this blue curve or shorter and wider like this green curve.
Now, that being said, let's talk about one of the most important conclusions we can draw from a normal
distribution, and that is breaking up the area under the curve into one, two and three standard deviations
around the mean.
So here we have a normal curve, we have the mean here at MU, and we've separated the area under the
curve based on standard deviation.
So this value right here, Mu plus sigma means the mean plus one standard deviation, whereas mu plus
two sigma here means the mean plus two standard deviations.
So here for this green curve, we said that the mean was here at five and that the standard deviation
was three.
So one standard deviation above the mean has to be five plus three or eight.
So we see that here at eight and then one standard deviation below, the mean has to be five minus three
or over here at two.
And so we can see for this green curve that we have the mean here and then this is one standard deviation
below the mean and one standard deviation above the mean.
Contrast that with this blue curve here where we see the mean at the same value of five.
And then we said that standard deviation was equal to one, which means one standard deviation above
the mean of five has to be at six here and one standard deviation below the mean.
It has to be at five minus one or four right here.
So we have four and six and we can say that for the blue curve, an interval of one standard deviation
around the mean ranges from 4 to 6.
Whereas for the green curve, an interval of one standard deviation around the mean ranges from 2 to
8, a much wider interval for one standard deviation around the mean, which of course makes sense because
we know that for the green curve the data is more spread out, variance and standard deviation are larger.
That measure of spread is larger, the data is more spread out away from the mean, whereas for the
blue curve, the data is more tightly clustered around the mean.
We see that smaller standard deviation.
So for the normal distribution, we're particularly interested in these intervals that define one standard
deviation around the mean between mu minus sigma and mu plus sigma, two standard deviations around
the mean between mu minus two sigma and mu plus two sigma and three standard deviations around the mean
between U minus three sigma and U plus three sigma.
Because what we know to always be true for normally distributed data is that 68% of the data will always
fall within one standard deviation of the mean.
And because the normal distribution is always symmetric around the mean, that means 34% of the data.
Or we could also say 34% of the area under the probability density function under this curve, or 34%
of the probability has to fall between one standard deviation below the mean and the mean, And 34%
of the probability or the area under the curve has to fall between the mean and one standard deviation
above the mean.
We also know that for every normal distribution, about 95% of the area under the curve will fall within
two standard deviations of the mean, and that about 99.7% of the area under the curve or the probability
will fall within three standard deviations of the mean.
These numbers, this idea, these percentages are called the empirical rule or more blatantly, the
6895, 99.7 rule given by these values 68, 95, 99.7.
But the empirical rule is what tells us that for every normal distribution, for every set of normally
distributed data, for every probability density function, representing a normal curve, any time we
have this bell shaped curve that is normally distributed, then we can always break up the area under
the curve based on these percentages, which allows us to answer all kinds of probability questions.
For instance, with this information alone, if we know that our normally distributed data has a mean
of five and a standard deviation of three, going back to this curve that we started with here and we
want to answer the question, what is the probability that a randomly selected value in our data set
will fall between five and eight or we'll have a value between five and eight?
Well, we know that the mean is five, and that the boundary here of one standard deviation above the
mean is eight.
So the probability that one randomly selected data point is going to fall between five and eight.
That's just.
This particular section of the normal distribution.
And we know that 34% of the area under the curve falls within this interval.
And so we know right away that there's a 34% chance that a randomly selected value from our data set
will take on a value between five and eight.
And of course, we can use these percentages in an infinite number of other ways.
So take this same data set.
What's the probability that we get a value less than five?
Well, five is the mean.
And in a normally distributed data set, 50% of the data is below the mean, 50% of the data is above
the mean.
So the probability of getting a value less than five is 50%.
The probability of getting a value greater than five is also 50%.
The probability of getting a value less than 11.
For this normal curve here the mean is five.
The standard deviation is three.
So one standard deviation above the mean is five plus three equals eight.
Two standard deviations above the mean is eight plus three equals 11.
So 11 is two standard deviations above the mean.
11 puts us right here at two standard deviations above the mean.
And so to find the probability that we get a value less than 11 would just mean that we add up all of
these probabilities, starting with 13.5 and then 3430 for 13.5 and 2.35.
Or how about the probability that we get a value outside three standard deviations around the mean?
Well, we know that within three standard deviations of the mean, we have 99.7% of the area under the
curve, which means the area outside of three standard deviations is 0.3%.
If we divide that by two to put half of that area into each tail, we see these values right here.
So instead of the total area in these two tails being 0.3% in this upper tail here we have 0.15%.
And in the lower tail here, we have 0.15% of the area outside of three standard deviations away from
the mean.
And that 0.15% captures all of the area under the curve infinitely.
Out to the left here and then infinitely out to the right here.
So within three standard deviations of the mean, we have 99.7% of the area, 99.7% of the probability.
This 0.15% in each tail captures everything else As far as this tail goes out to the left or as far
as this tail goes out to the right, everything outside of three standard deviations.
So then, for instance, if we wanted to say what's the probability that we get a value less than 11
going back to this normal curve here with a mean of five and a standard deviation of three?
Well, if our mean is five and then one standard deviation above the mean is five plus three or eight,
then two standard deviations.
Above.
The mean is five plus six or 11 or eight plus three equals 11.
So we know that 11 for this curve puts us at two standard deviations above the mean, which is right
here.
So the probability that we get a value less than 11.
All we have to do is add up all these percentages 0.15, 2.35, 13.5, 34, 34 and 13.5.
Or to take the easier route since probability always sums to 100% or one, we can just take 100% and
we can subtract from that this 2.35.
Percent value and this 0.15% value and we get a result of 97.5%, which tells us that we have a 97.5%
chance of finding a value less than 11.
Given this normally distributed curve with these properties a mean of five, a standard deviation of
three, it's this idea of the empirical rule and these standardized percentages, because the normal
distribution is a standard curve that's going to allow us, as we continue on here, to answer all kinds
of probability questions and then even start doing some statistical analysis on normally distributed
data, which ultimately leads us to an idea about significance, being able to start with a hypothesis,
run an experiment or collect data, and then determine the significance of the result of that data to
determine how unusual it is to find the result that we did.
We'll talk all about that when we get to hypothesis testing.
But the idea here is that this normal distribution we see over and over and over again in real life
and almost more importantly, even when we collect data and we determine that it's not normally distributed,
we'll learn about a way that we can sort of change that non normal data into normally distributed data
so that we're able to use this normal distribution and all of these probability figures that go along
with it.
And so because we'll be using this normal distribution over and over and over again for normally distributed
data and even non normally distributed data, it's critical that we understand this idea of this symmetric
bell shaped normal distribution, where the empirical rule can give us the likelihood that a particular
data point falls within some certain standard deviation interval around the mean.