All we want to do here is learn how to convert from the normal distribution to what's called the standard
normal distribution.
So previously we looked at a normal distribution and we said that the normal distribution was this symmetric
bell shaped curve that had its center, both its mean and its median.
At the center of the distribution here where the mean is mu.
We said also that the normal distribution followed the empirical rule which told us the percentage of
area that we could expect to find under each section of the curve where these sections here are broken
up based on the standard deviation of the data.
So this blue area here represents one standard deviation around the mean.
So from the mean minus one standard deviation on the left to the mean plus one standard deviation on
the right.
If we add on to that and include this red area that represents the interval from the mean minus two
standard deviations on the left all the way over here to the mean plus two standard deviations on the
right.
And then we have our interval for three standard deviations around the mean as well.
But with a normal distribution as we saw before, the mean and the standard deviation can take on any
values.
In fact, we said that the normal distribution could always be identified as capital N and then mean
and variance.
So for instance, if we had a mean of five.
And a standard deviation of three.
We could indicate that normal distribution as five and then nine.
Because standard deviation is sigma, variance is sigma squared.
So if our standard deviation is three, then our variance is three squared or nine.
And in fact, we looked at that exact normal distribution.
But the standard normal distribution is the special normal distribution where the mean is always zero
and the standard deviation is always one.
And we indicate the standard normal distribution with the capital letter Z instead of just any normal
distribution with the capital letter N.
So if we see capital Z, that's indicating the standard normal distribution.
And of course if we were writing this as a normal distribution, we could write it as capital N of zero
comma one, because in the standard normal distribution, the mean is zero, the standard deviation
is one, therefore the variance is one squared or one.
So the standard normal distribution is just equal to the normal distribution that we've looked at before.
But with this particular mean and standard deviation, which means that if we're sketching the standard
normal distribution here, we have to indicate that the mean is at zero.
And then by definition, the lower boundary of one standard deviation around the mean is negative one,
the upper boundary is positive one, and then we have negative two and negative three and positive two
and positive three for the boundaries around the two standard deviations around the mean interval and
three standard deviations around the mean interval.
So when we have a standard normal distribution, then these boundaries are always negative three, negative
two, negative one 0123.
Whereas in this example here for the normal distribution with mean five and standard deviation three,
those values would have been a mean of five.
And then with a standard deviation of three, we would have had eight, 11 and 14, and then here two
negative one and negative four.
Now what we want to understand at this point is how we can take any normal distribution and convert
it into the standard normal distribution.
In other words, to standardize it.
If we could have a way to convert any normal distribution into this super standardized, normal distribution
with this mean of zero and standard deviation of one, that would certainly make comparing normal distributions
easier.
It would make identifying values more clear.
We could make our reporting more clear.
It would just give us a good standard from which to start.
Instead of starting with something like this, this normal distribution we looked at earlier with mean
five and standard deviation three, where we have all of these different numbers and it's much harder
to see that this is a normal distribution.
And of course, as you may expect, we already have a way to convert from a normal distribution into
a standard, normal distribution.
And it goes back to what we looked at earlier when we learned about transforming random variables.
Remember that we talked about how to apply a shift or a scale to a random variable, and that's all
we're going to do here.
We're going to start with our normal distribution.
MN And we're going to apply a shift and then a scale in order to transform our normal distribution into
the standard, normal distribution.
So if we start with a normal distribution and instead of representing that random variable with N,
let's say we're representing it with X, then the first thing we would do is apply a shift to X where
we subtract out the mean.
Now let's talk about what this looks like visually.
So this is in fact a sketch of the normal distribution with mean five and standard deviation three.
So what we're starting with here is the normal distribution of the random variable here with mean five
and standard deviation three or variance nine.
This is what it's probability distribution looks like.
Now if we apply a shift by subtracting out the mean from every data point in our dataset that's making
this distribution, it's going to move every data point to the left by five units.
The mean is five.
So when we subtract out the mean, the graph will shift and the curve moves over to the left.
And now we have a distribution that is centered at zero that has a mean of zero.
So we moved the mean from five here to zero here by applying this shift to all of the data points in
our dataset.
So now we have the mean of the standard normal distribution, which is that mean of zero, but we still
have a standard deviation of three.
We still have this wide stretched curve.
So this curve we have right now, we started with the normally distributed random variable with mean
five, standard deviation three.
Now this curve that we're showing is the normally distributed variable with mean zero and standard deviation.
Three.
So we have the first piece of getting this over to a standard normal distribution.
Now all we have to do is apply a scale, and here's what that scale looks like.
We will divide every data point in the data set by the standard deviation.
When we do that, when we divide by the standard deviation, that means every data point in the set
is going to get pulled in closer to the mean in this case.
And the resulting curve looks like this and this.
Now, this final curve, after both the shift and the scale, is the normally distributed random variable
with mean zero and standard deviation one, which of course we know to be equal to the standard normal
distribution.
So that's really all that we want to understand how we transform one normal distribution into the standard,
normal distribution, regardless of the mean and standard deviation that we started with.
And going forward, this formula here after we've applied the shift and the scale is going to be our
next point of focus.
So we want to imagine that we started with the random variable x, and after these two transformations
we transformed it into the random variable Z.
And now if we rework this formula a little bit, if we switch it around and consolidate this side here
we get Z is equal to X minus the mean divided by the standard deviation.
And it's this formula that we're going to be focusing on going forward as we talk about what we call
the Z scores that are associated with the standard normal distribution.