We're devoting this entire section to talking about the normal distribution because it's such an important
distribution for modeling real world phenomena and because it's so important because it comes up so
often, we're going to break it down into several pieces here.
We just want to talk about the mean variance and standard deviation that are associated with a normal
distribution.
So as a reminder, a normal distribution is a distribution whose probability density function looks
like this.
So if the probability density function is F of x, then this curve here is modeled by this function
F of x because of the shape of the normal distribution.
We often call it a bell curve as well, although there are other distributions that have bell shapes.
So this particular kind of bell shape is not exclusive to a normal distribution, but all normal distributions
do have this kind of symmetrical bell shaped curve or bell shaped probability density function, which
is centered at the mean of the data, which if we're talking about a population, we indicate with MU,
if we're talking about a sample, we indicate with x bar.
And that's right at this point right here, the center of the normal distribution.
Now when we're talking about a normal distribution, we have to keep in mind that we could still be
referring to either a population or a sample.
Remember that a population is the entire group or set that we're studying, whereas a sample is a small
subsection of that population.
And we distinguish between these two things because let's say, for instance, we're interested in surveying
the entire population of the state of California.
We know right away it's going to be very difficult, if not impossible, to collect survey responses
from every single person, every single citizen of the state.
And so instead of attempting to ask every person in the state the same survey question or to gather
data from every person in the state, the entire population, instead we'll take a small sample.
The goal, of course, is to make sure that our sample is representative of the population, but that's
an entirely different topic.
Figuring out how to take a sample that isn't biased but instead is representative of the population.
Now, for a normal distribution, our formulas for mean variance and standard deviation will be different
depending on whether we have data from the entire population or just from the sample.
Of course, keep in mind that normally we'll have sample data because practically speaking for real
world problems, it is usually difficult or impossible to collect data for the entire population.
But to distinguish between these two groups, we want to indicate the size of the population with capital
N and the size of the sample with lowercase n.
So if you see capital N, that means we're talking about the number of subjects in the entire population.
So if we're talking about the state of California, this capital N would represent the number of people
in the entire state.
If we see lowercase n, it means that we've taken a smaller sample of the population and the number
of people in our sample is given by lowercase n.
So let's say based on what we're studying, not related to the California example, our population is
10,000.
Well, capital N would equal 10,000.
We might take a sample of, let's say 1200 people and then lowercase n would be equal to 1200.
So these are the sizes of the population in the sample capital n and lowercase n, And then the mean
of the population is given by this formula here, whereas the sample is given by this formula.
Notice that they are identical, except for two things.
First, in the population mean formula, we're dividing by capital N, whereas in the sample mean formula,
we're dividing by lowercase n, And of course that makes sense based on the fact that we're using capital
n and lowercase n to represent the size of our population and sample respectively.
And then the only other difference is the notation that we use to indicate the mean, which we talked
about earlier up here as the center of the normal distribution.
The population mean, we indicate with this Greek letter mu and the sample mean we indicate with x bar
this x with a bar over it.
But other than that, the formulas are identical.
So all these formulas are telling us is that to define the mean, we are summing up all of the possible
x sub values, all of the values that are variable takes on and after we sum them all up, then we divide
by either the number in our population.
If we're finding population mean or the number in our sample, if we're finding sample mean.
So add up all the data points, divide by the number of data points we have.
That's all we're doing to calculate the mean of the normal distribution.
Now for the variance, this should look familiar because we've already looked at mean variance and standard
deviation formulas.
But for the normal distribution, just to confirm here, all we're doing with the variance is.
Taking all of the data points that we have, all of the sub values that we have in our data set.
And for each one we subtract the mean here.
So notice that the formulas for population in sample are again almost identical.
All we're doing is taking the individual x abi values in our data set and that goes for population and
sample and then we subtract the mean.
And of course if that's the population we're subtracting MU if it's the sample, we're subtracting X
bar.
But in either case we're just subtracting the mean and that gives us the deviation for each data point.
So this is the deviation of the individual data point x by when we square that value, we now have a
squared deviation or a squared difference.
And then this summation notation just tells us to sum up all the squared deviations.
So once we expand out to this part of the formula here, this is just the sum of all the squared deviations.
And again, notice that up to that point our formulas are identical, except for the mean being represented
by MI or X bar.
And then in the summation notation for population, we're summing up to capital N for the sample, we're
summing up to lowercase N, So really no difference at all, practically speaking, between the two
formulas.
Up to this point, we're just taking the sum of all of the squared deviations.
But then for population, we divide by the number in our population capital.
N, But for the sample, you might think that we would divide by lowercase n the number of subjects
in the sample.
But in fact, and this is the most interesting part of all of our formulas for mean variance in standard
deviation for a population in a sample.
This particular piece right here is the most interesting part of our formulas, and the reason is because
we would think it's n, but it's actually n minus one.
And that's because when we take a sample from a population, we inherently introduce some bias.
It's virtually impossible for our sample to perfectly represent our population for us to be able to
scale up our sample data to perfectly match our population.
For example, maybe we're working with a population of people and let's say the population is 1000 people,
and we'll even say that exactly 500 of them are women and 500 of them are men.
We might take a sample of 100 people, and even though we randomly sample from the population, meaning
that we are randomly, blindly choosing one person at a time from the population, putting them back
in the population, and then pulling out another person.
And we're collecting a sample of 100 people from that population of 500 women and 500 men.
When we sample that way and we collect 100 people, we might get a sample that's, let's say 52 women
and 48 men.
That's not going to scale up perfectly to 500 women and 500 men.
Our sample is going to be a little bit different in terms of its representation of the population.
We are just hoping and implementing some good sampling techniques to try to make sure that the sample
is as representative of the population as we can possibly make it.
But no matter what we do, we're likely going to introduce bias.
And so because we introduce some bias when we sample from the population, it turns out that because
of that bias, using this end minus one figure to calculate variance actually helps us undo a little
bit of that bias as opposed to just using lowercase n the number of subjects in our sample.
So we use this n minus one value to get a better approximation to get rid of some of the bias we introduced
by sampling instead of using the full population.
So when we calculate variance for a sample, we have to remember to divide by n minus one instead of
just lowercase n the number of subjects in our sample.
So if we take a sample of 100 people, we would be dividing here by 99 instead of dividing by 100.
So those are our population and sample variance formulas.
And then, as you already know, standard deviation will always just be the square root of variance.
Now, if we want to see these formulas in action, of course the easiest way to do this is going to
be to use a calculator or a computer to help us make these calculations, especially as our dataset
gets larger.
Because you can see here, let's work with a data set that I've already created.
This data set only has ten data points in it and already look at all of the calculations we're doing,
much less if we have a data set of 1000 subjects or even more, these calculations quickly become impossible
to do by hand.
But here's what we've done here to calculate mean variance and standard deviation for this particular
data set.
Let's walk through this step by step.
So what we're going to say is that maybe we're a distribution company.
We're running a.
Warehouse.
And we want to get information about the age of the employees that work in the warehouse.
So what we're going to do is we're going to take a sample of ten employees.
So we're going to say lowercase n is equal to ten.
We're going to be using a sample instead of surveying the entire population.
Maybe we actually have 200,000 warehouse workers and we're not going to be able to ask every one of
them for their age.
So instead, we're going to take a sample of ten employees.
And all we're doing here, we're not talking about at this point whether this is a good sample.
We'll talk about sampling techniques and representative samples later.
All we're talking about now is how to calculate mean variance and standard deviation for this sample.
We're using age in this example only because age is as long as our population is large enough, is going
to be normally distributed around some mean.
So we're going to say here that our sample size is lowercase n equal to ten.
We surveyed ten employees, and this second column here of our table are all of the responses we got.
So we asked ten warehouse employees their age and we got all ten ages here.
And then to calculate sample mean.
So we're calculating here X bar we found that x bar the sample mean was equal to 43.2.
And we find that by taking the sum of all of these ages and then dividing by N equals ten.
So we would take here to get x bar the sample mean.
This is how we calculated 43.2 we would take 43 plus 35 plus 38.
So 43 plus 35 plus 38.
And we would continue adding until we got to the very last age here 40.
And then we divide that entire sum by the sample size and equals ten.
And we see that here in our sample mean formula.
We sum up all of our data points, all of our x sub data points, and then we divide by the sample size
and equals ten.
And the result there is a mean of 43.2 for the age of those ten employees.
Now, to find variance for our sample, we take each one of our x sub values and we subtract the mean
x bar.
So that's what we're doing here in this column.
We're finding all of the deviations.
So we're taking 43 and subtracting 43.2 to get negative point two.
We're taking 35 and subtracting 43.2 to get -8.2 all the way down until the last data point.
And so this second column here is giving us this x sub minus x bar value for each data point.
Then in this last column here, we're squaring each of those deviations.
So -0.2 squared is a positive 0.0 for -8.2 when we square it is a positive 67.24.
So we square all those values.
Now we have all of our individual squared deviations.
This part of the formula, then our summation notation here tells us add up all of those squared deviations.
So we have all of our squared deviations and then we add them all together to find this sum.
Now we have the sum of all of our squared deviations, and then all we have to do to find variance is
divide this sum of all the squared deviations by and remember and minus one because we have a sample,
not a population.
And when we find sample variance, we have to divide by n minus one to undo some of that bias.
So here this value for variance is not 1,007.6 divided by N equals ten, it's 1,007.6 divided by ten
minus one or divide it by nine.
So when we divide the sum of our squared deviations by n minus one equals nine instead of n equals ten,
the result we get is 111.96.
And so this is our variance, 111.96 and that's an approximate value because we're rounding there to
the nearest two decimal places.
So that's our approximate variance.
And then of course, to find standard deviation, we just take the square root of variance.
So standard deviation will be equal to the square root of variance, which in our case is approximately
the square root of 111.96, which is approximately.
10.58.
And so our standard deviation for the sample is approximately 10.58, just over ten and a half.
Remember and we talked about this before, that the mean is this center of the data, The standard deviation
is the figure that we'll use as a measure of spread mean is a measure of central tendency.
Standard deviation is a measure of spread.
So this figure tells us how much this data is spread out around the mean.
Remember that if our standard deviation is smaller, it tells us that our data is more tightly clustered
around the mean, which means all the data gets sucked in closer to the mean.
And our normal distribution might look something more like this with a taller peak and smaller tails
where all this data is pushed in closer to this center.
The mean.
Whereas if our standard deviation is larger, it tells us that our data is pushed further away from
the mean.
It's more spread out.
This measure of spread is larger, and so our normal distribution might look something a little bit
more like this, where we have fatter tails and the distribution is shorter, the data is pushed out
further away from the mean because of that larger standard deviation.
And we'll talk more later about how we use and interpret mean variance in standard deviation.
The primary goal here is just to understand how to calculate these three values for a normal distribution
for both a population and a sample.