>> This video is about the sample variance
and the sample standard deviation.
Variance and standard deviation is about the spread of the data.
The more the data spread out the larger the variance
or the larger the standard deviation.
Standard deviation is just the square root of the variance.
And we're going to do an example here
that will help you understand the concept of the variance
and the standard deviation.
So, let's define the sample variance.
Sample variance is the average which is the keyword
of the squared deviations, another keyword,
from the sample mean, mean is another important word,
another keyword.
So, the sample variance is represented as esquire
and that's equal to the sum of the squared deviations.
From i equals 1 to n divided by n take away 1.
XI we know as the points and x-bar you know as the mean
and so the deviation here is squared,
the squared deviations from the mean.
The big piece here is that a lot of students don't know how
to calculate this piece right here and that's
where we're going to focus our attention by doing the table.
But first we need some data points and we're going
to write a table with some columns
to help us do this problem.
And you notice that to do this column right here we need the
mean, which is not too hard to do.
That's going to be x-bar which is equal to the sum
of the data points divided by the number of data points,
in this case it's fine.
So, we add the top 20 divided by 5 and that gives you 4.
So, the next step is to create this column right here
for the deviations for the mean, it's not squared
yet it's just the deviation.
So, we're going to take four which is our data point
and we're going to subtract it from our mean
which is four as well.
When we subtract it, it gives us a zero.
So, the next one is take your data point which is three
and subtract it from the mean and that gives you negative 1.
Now let's do the rest, 7 take away 4 is 3, 5 take away 4 is 1,
and 1 take away 4 is negative 3.
Okay, so let's create another column and now the deviations
for the mean let's square them.
So, we're going to take each answer that we got
from the deviations from the sample mean
and we're going to square it.
So, the first one is zero squared,
which will give you zero.
The next one is going to be negative 1 squared,
which will give you 1.
One key thing about doing this column is
that all the answers are positive
because whatever deviation you get it's going to be squared.
Another key thing is do these by mind, you know write it
out if you want to, don't use a calculator just do it.
Negative 1 times negative 1 is going
to be negative times negative is positive 1 times 1 is 1.
So, do that and you won't make a mistake there.
Let's try the next one, 3 squared gives you 9,
1 squared equals 1 and the next one is negative 3 squared.
Again, negative 3 times negative 3,
negative times negative is positive, 3 times 3 is 9.
So again, don't use your calculator use your mind you'll
be able to do it and that gives you 9.
Now let's find the sums of the columns, pretty straightforward.
What's the sum of all the data points,
that's going to give you 20?
What's the sum of the deviations from the mean,
add all that up it gives you zero?
Most of the time it'll give you zero,
it might be a little bit off if you have to round.
And the next column the squared deviations from the mean
and that's going to give you in this case 20.
All right, so the question is what is the sample variance?
Well that's of course the formula
and let's fill in those parts.
The sum of the squared deviations from the mean right,
we calculated that already is 20 so let's put that in there.
Divided by the data points,
how many data points do we have, five?
Subtract 1 from formula so this gives you 20 divided
by 4 and that gives you 5.
So, that is the sample variance.
You also want to calculate the sample standard deviation.
To do that it's not a problem.
And that's going to be represented as s
and we take the square root of the variance which is s squared.
So, s is a sample standard deviation
and s squared is the variance.
And therefore, doing this problem that means we want
to find the sample standard deviation we're going to have
to plug in 5, take the square root of 5
and that gives you 2.4.
All right, now we're going to do the sample standard deviation.
And let's answer this question.
The question is what is one sample standard deviation
from the mean?
What we want to do is we want to know some distance on the right
of the mean and on the left of the mean that is equal
to one standard deviation on the right
and one standard deviation to the left.
But because it's on the left we'll say minus one
standard deviation.
And we want to know where that distance is,
where is the distance, where is it located at on this axis.
So, to do that we have a formula that we can follow
and the formula is x-bar plus or minus s. And let's write it
out again, x-bar plus minus s
which tells me it's x-bar plus s that's the first formula.
And x-bar minus 2 which is the next formula.
And so, let's plug it in.
We know what x-bar is 4 plus s which is 2.24,
we add that that's going to give us 6.24.
And because it's two equations we've got
to find also the other equation on the left,
4 take away 2.24 that's going to give me 1.76.
All right, so what is that distance,
6.24 and 1.76 these are distances
that are from the mean.
So, let's draw the first one and that's --
we start off, here let's do that again.
We start off at 6.24, somewhere around here we start
and then we draw 2 mean that's one standard deviation
and we'll write it as 1 s. Well on the other side 1.76,
somewhere around here that's where we're going to start
and we're going to draw it all the way towards the mean.
Because it's on the left side it's negative one
standard deviation.
A little confusing yes, but that is the convention,
the convention is anything to the right
of the mean is positive, anything to the left
of the mean is negative.
All right, let's do two sample standard deviations
from the mean.
In this case, our formula is going to change a bit
because we're dealing with two sample standard deviations.
So, we're going to do that x-bar is plus or minus 2 s. So,
that's going to be pretty easy.
So, x-bar plus or minus 2 s is 4 plus 2 times 2.24 right,
we do the math we get 8.48 and also
on the left side 4 take away 2 times 2.24
that gives you negative .48.
So, let's draw the line now, the distance 8.48 to the mean
and negative .48 to the mean as well.
So, 8.48 that's around here somewhere so the 9 is going
to go that way so let's draw that.
So, that's two standard deviations.
And the next line right, we have .48 that's around here somewhere
and it's going to go that way so let's draw that one.
So, from negative .48 all the way
to the mean that's negative two standard deviations.
So here it is.