So we left off talking about the standard normal distribution and this formula here for Z, where we
say that Z is equal to X minus MU, divided by sigma, where X represents the random variable that we
started with Z is the random variable specifically associated with the standard normal distribution
MU is the mean and sigma is the standard deviation.
And the point that we want to make now is specifically that if our random variable x is normally distributed,
we can always use this formula to convert that normal distribution into the standard normal distribution
associated with Z.
So if we pull back up just a normal distribution, we said that sometimes we have normal distributions
like this with this bell shape, but we can have normal distributions that are a little shorter and
wider.
So the distribution might be shorter than this one, but we can also have normal distributions that
are taller and narrower than this one.
They're all normally distributed, but they just have different measures of spread, they have different
standard deviations.
They're still normal, but some are more spread out and some are less spread out, whereas the standard
normal distribution Z always has that same mean of zero and standard deviation of one.
So that measure of spread for the standard, normal distribution is always the same.
Standard deviation is one, whereas we can have infinitely many normal distributions represented by
X with different measures of spread with different standard deviations.
The point here is that regardless of the normal distribution that we have, regardless of the distribution
of X with its own mean and standard deviation, we want to use this formula, which, if you remember
from the last lecture, uses a shift of MU and a scaling of sigma to change the random variable x into
the random variable z.
We always want to use this formula to convert all of the values that make up the data.
Set X into their corresponding values in the data set Z, and we can see this formula working in action.
Remember that throughout the last few lectures we've been looking at a normal distribution that has
a mean of five and a standard deviation of three.
This normal distribution is obviously not the standard normal distribution, because the standard normal
distribution has a mean of zero and a standard deviation of one.
And this normal distribution has a mean of five and a standard deviation of three.
So we're dealing with the normal distribution, but it is not the standard, normal distribution.
So these two pieces of information here come from our random variable x, which is normally distributed,
and we want to convert X into Z.
We want to change this distribution into the standard normal distribution Z.
And you can see that if we just take a few test values here, let's say that the random variable X takes
on a value of five.
Well, if we plug in five for x five from you and three for Sigma, what we get is Z equal to five minus
five divided by three or zero.
So the Z score associated with x equals five.
We'll put a little five subscript here to indicate that x was equal to five.
The z score associated with x equals five is zero.
Let's look at what happens if we set x equal to eight.
So we say z for the x value eight is equal to we would get eight minus five divided by three or three
divided by three is equal to one.
If we pick, let's say x equal to two.
We get two minus five divided by three is equal to negative one.
And if we pick maybe x equals 6.5.
We get 6.5 minus five divided by three is equal to 1.5 divided by three or one half or 0.5.
Now, here's what these values are actually telling us.
The value we get for Z is giving us the distance of whatever X value we picked five 8 to 6.5.
It's giving us the distance of that particular value of the random variable x away from the mean in
terms of standard deviations.
So when we picked x is equal to five, the mean of the random variable x is five.
So the distance between the x value five and its mean is of course zero.
So the z score we get is zero because the distance between the x value we picked and the mean is zero.
So we get a Z score of zero, but if we pick X is equal to eight, the distance between eight and the
mean five is three.
We see that here, eight minus five in the numerator is three.
So we could say that the distance between x equals eight and its mean is three.
But we always want to give the distance in terms of standard deviations and that's why we divide by
the standard deviation sigma.
So when we take three divided by three, the result is one.
And so this value tells us that the distance of eight away from the mean five in terms of standard deviations
is one.
In other words, X equals eight is one standard deviation away from the mean, more specifically one
standard deviation above the mean because we got a value of positive one.
But when we pick x is equal to two and we calculate the associated Z score, we say that the distance
between x equals two and its mean x equals five is two minus five or negative three.
So the distance between the specific value of x that we chose and its own mean is negative three that
negative three value.
The fact that it's negative tells us that the specific value of x that we picked is below the mean.
It is less than the mean, which of course we see x equals two is less than x equals five the mean.
So we get a distance of negative three and then when we divide by the standard deviation of three,
the result we get is negative one.
And this result tells us that x equals two is one standard deviation below the mean.
So it's one standard deviation away from the mean.
The negative sign tells us one standard deviation below the mean, whereas over here the positive one
tells us one standard deviation away from the mean and specifically one standard deviation above the
mean.
And then here we see x equals 6.5.
The distance away from the mean five is one and a half, right?
6.5 minus five is 1.5 or one and one half.
So our distance away from the mean for this particular value of x is 1.5.
But what is that distance in terms of standard deviations?
Well, it's half of a standard deviation because one standard deviation is three.
So what is 1.5 of three?
1.5 is half of three.
And so the result that we get is 0.5 or one half.
6.5 is half of a standard deviation away from the mean.
And so this standardized Z score is 0.5 or half of a standard deviation above the mean.
That is what we're calculating These values here that we're calculating are Z scores.
And the Z score always tells us how many standard deviations we are away from the mean when we choose
a particular value of our random variable.
If we get a positive value for the Z score, it means the particular value we chose is above the mean.
If we get a negative value, it means the value we chose is below the mean.
If we get zero, it means the value we chose is exactly at the mean, it is the mean itself.
Now we don't call the distribution of Z the standard normal distribution for nothing.
It really is standardized, so much so that we actually have a full table of values for every different
Z score that we can possibly calculate.
And here's what that big table looks like.
Now, there's a lot going on here, but it's actually pretty simple when we break it down, What we
see in this whole left side of the table here is this first column of Z scores.
And these values are all negative.
You see that in the first row.
We start with -3.4 and we go all the way down here to the bottom where we see -0.1 and then 0.0.
So these are all negative Z scores, which we already said we find when we pick a particular value that
is below the mean.
We saw that here when we calculated the Z score for X equals two, whereas this whole right hand side
of the table we see here in this column, the Z scores are all positive.
They start at 0.0 and move all the way up to 3.4.
So you can almost think about the normal distribution.
And if we're starting at the left edge of the normal distribution, we are starting here with this most
negative value, -3.4.
And as we move from left to right along the.
Normal distribution.
We move this way in our Z table until we get all the way down here to the bottom at 0.0, where we reach
the halfway point of the normal distribution or the mean of the normal distribution.
Because remember, a Z score of zero is associated with the mean.
We saw that here and we see that this Z score of zero is giving us exactly 0.5.
And that's because 50% of the area under the curve of the normal distribution is to the left of this
Z score.
And we'll talk more about that in a second.
But we move all the way down through the table here, all the way through these negative Z scores,
from the most negative Z score to a just slightly negative Z score of -0.1.
We hit this 0.0, which is right at the mean or right at the middle.
And then we continue on through this right side of the table starting at 0.0.
The mean.
So you can see how these two values match between the tables because they're both at 0.0.
And then we continue on through positive Z scores, starting with a slightly positive Z score of 0.1
until we get to an extremely positive Z score of 3.4.
And of course, up here we're at the far right edge of the standard normal distribution.
The idea here with these Z tables, the value that we're getting out of the body of the Z table is the
percentage of area underneath the curve that falls to the left of the Z score that we look up.
So we already said here that for a Z score equal to zero, we could look in either side of the table
and we see this 0.0 value.
So we go to 0.0 that takes us out to the first decimal place.
But then we have to look across the top row of the table to get the second decimal place for a Z score
of exactly zero.
We want to stay in this first column here to get a true Z score of 0.00.
And so we go to the last row, this first column, and we see here 0.5.
That means that 50% of the data, 50% of the area under the standard normal distribution lies to the
left of Z equals zero.
If we want to look up a Z score of one, well, this is a positive one here.
So we're going to be in this right hand side of the table.
We're looking for a positive one.
Well, we find that at 1.00.
And so we see this value right here, which is 0.8413.
That tells us that about 84% of the data or 84.13% of the area under the standard normal distribution
lies to the left of Z equals one.
Now, if we look for Z equals negative one, because we have a negative Z score, where in this left
side table here, if we go to Z equals -1.00, that puts us here in this first column at 0.1587.
This tells us that the area under the standard normal distribution to the left of negative one is about
15.87% of the total area.
Now, here's an interesting point to make note of.
Now that we have the Z score for both positive one and negative one, if we just sketch a super simple,
normal distribution, we'll say that it looks like this doesn't need to be perfect, but we have here
our.
Mean, and we just found the percentage of area below Z equals one and Z equals negative one.
So let's say that this right here is one standard deviation above the mean.
We'll say that's one and that then symmetrically.
This is one standard deviation below the mean or Z equals negative one.
What these values tell us is that 15.87% of the total area under this curve is going to fall to the
left of Z equals negative one.
So this area right here makes up 15.87% of the total area under the curve.
Another way to put this is that the probability that we find a value below one standard deviation below
the mean is 15.87%.
Or in the case of this normal distribution here, this random variable x, remember we found a z score
of negative one for this value x equals two, which means that the probability or the likelihood that
we get a value of X in our data set that is less than two is about 15.87%.
Now, remember, the standard normal distribution is symmetric, which means that if the likelihood
of us getting a value below one standard deviation is 15.87%, that means that finding a value above
one standard deviation.
So over here, the area that is symmetric to the one that we already sketched over here, the probability
of us finding a value above one standard deviation above the mean also has to be 15.87%.
And of course that makes sense because when we looked here in the Z table for the value associated with
a Z score of 1.00, we got 0.8413.
So we're saying that the probability that we find a value for X that's less than one standard deviation
above the mean is 84.13%.
So this whole area here, everything below one standard deviation above the mean or to the left of a
Z score of positive one.
This whole area is 84.13%.
So that is 84.13%.
And we're saying this value in blue, which was symmetric to the blue area that we sketched on the left,
that is 15.8, 7%.
So if we add to that 15.87%, notice that what we get here.
Is 100%.
Which, of course makes sense because the area under the distribution always has to be equal to one
or 100%.
And so when we pick some Z score like Z equals one, and we look at all the area to the left of Z equals
one, and all the area to the right of Z equals one, those two areas need to sum to one or those probabilities
need to sum to 100%.
They're complementary probabilities and we can see that with any two associated Z scores in this table.
Let's say for instance, we pick 1.53 that would give us this value right here, 0.9370.
And then if we go find -1.53, so we go over here to the negative side of the table, -1.53 gives us
this cell here, which we see is 0.0630.
And if we add 0.9370 to 0.0630, the result we get is one.
So the values we get from the Z table for opposite Z scores are always going to add to one which makes
sense given our understanding of the area under the probability distribution, always adding up to one.
We can also think about the values in the Z table as percentiles.
So if the value that we get from the Z table is let's take this value here 0.9370, that tells us that
the data point that was associated with a Z score of 1.53 is greater than 93.7% of the data points,
or it's in the 93.7 percentile of all of our data.
Or to say it another way, whatever value of the random variable x we used to arrive at a Z score of
1.53 is a value of X larger than 93.7% of the other values of X and less than 6.3%, which we see right
here of all of the other values of X.
Now, keep in mind too, that we can also use this Z table to work backwards.
So if we want to know, for instance, what the threshold is or the cutoff for the top 30% of our data,
well, top 30% means that we have to be in that 70th percentile.
100%, -30% is 70%.
We have to beat that 70% cutoff.
So if we come here to the positive side of the Z table and we locate the first value that is greater
than 0.7, we see that it's right here, this cell right here, 0.7019.
This is the first cell we see that is greater than 0.7.
Just to the left of it, we see the first value that is less than 0.7.
So we know that if we want to be within the top 30%, we'll need a Z score of at least 0.53 because
we get the 0.5 from the row header over here, 0.5 and we get the 0.03 from the column header up here.
So 0.5 and then 3.53 is the smallest Z score we can get that's associated with the top 30% of our data.
So if we choose a value of X and we calculate its Z score and we get a Z score that is greater than
or equal to 0.53, we know based on the table here that we're going to be in the top 30% of our data.
So that's the idea of a Z score, the formula we use to calculate it, what it's actually telling us
how to look up the Z scores that we find in a Z table, what the information in the Z table actually
tells us, and how to work backwards from the Z table to the Z score.
We will use these tables all the time and a quick search online for Z table will pull these up right
away.
They're so standard that you'll be able to find the Z table everywhere.
But this is what we're going to be doing all the time going forward.
We're going to have some random variable, we'll call it X, we're going to know its mean and it's standard
deviation.
But instead of using the normal distribution that's associated with X, which might be shorter and wider
or taller and narrower than the standard normal distribution that represents Z, we're going to convert
all values of X into values of Z to essentially find the associated value of Z for every value of X,
We'll calculate here a Z score for any value of X that we need.
That'll give us the value of Z that's associated with a particular value of X that we're interested
in.
In other words, it'll give us our Z score for that particular value of X, And once we have that Z
score, then we have these Z tables available to us as a tool which will allow us to quickly look up
all of these probability percentile threshold values that are associated with any specific Z score that
we've calculated.