Instructor: So, hello, students.
We are going to now go to Problem 1B.
And here, we are going to learn how to calculate
the standard deviation.
So again, first thing I suggest my students to do
is write down the formula.
So what is the formula to calculate the standard deviation?
It is basically,
looks a little intimidating, but it's not,
and I'll show you how to do this in a very simple way.
So, this is our formula to calculate standard deviation.
The best way to do this is to go step by step,
and I'll show you how you can even double check
whether you're doing it correctly or not in the process.
So, I'm going to put a table here.
X sub i is basically our X values.
The middle column here is X sub i minus X bar,
which is our sample mean, which we have already calculated.
And the third column is X sub i minus X bar
and whole squared.
So basically, we are going to whole square the difference.
So again, here, we are going to write the numbers,
the ordered set, which I showed you before,
147, 175,
180, 185,
194, 196,
223, 225,
and 240.
So, the middle column is basically the difference between
this number and the mean.
So for the first one, it will be 49.1.
Second one is negative 21.1.
Third one is 16.1, again, negative.
Fourth one is minus 11.1.
Fifth one is, again, negative 2.1.
Sixth one is negative 0.1.
Now this is going to become positive 26.9.
And this will also be negative 28.9.
And the last one is also gonna be a positive number,
which is 43.9.
Now what I want you to do
is double check whether you are doing this correctly or not,
is to add all the positive and negative values
and see if you come up with zero.
So we are gonna add these numbers,
and add all of these numbers.
So when we do that, what we come up with is negative 99.6
and a positive 99.1.
So, this is what?
It still is a negative 0.5, which is very close to zero,
and the only reason we are getting a negative 0.5
is because we have a decimal here,
if we had a whole number,
this would end up actually being zero,
and that tells you that you have done this correctly.
Then the third column basically is
to whole square the difference.
And when we do that, these are the values we get.
And again, anytime we whole square a negative number,
it always becomes positive.
So what we have done
is we have taken the differences
and we have whole squared them,
which means multiplying the number by itself,
and these are the values we have obtained.
Okay.
I already performed the calculation,
so it would not take a humongous amount of time
for us to go through this,
but again, I strongly encourage you
to do the calculation yourself,
just to make sure you understand the step by step process.
And of course, if you have any questions,
please let me know.
So what we are going to do
is right now sum up the third column,
and that will be 6,728.9.
So as you can see right now,
we have calculated our numerator,
which is kind of the most, I would say,
intimidating part of this formula.
And the sample size is still what?
Nine.
So we deduct one from nine, we get eight.
So we put that here, 6,728.9,
we divide that by 8.
We get,
there's a decimal here,
and when we do the square root,
we get 29.0019.
Given the decimals,
the situation here, if we just round it up,
we will get 29,
because it is too far out to really use the decimals here,
because it would be what?
29.002, so we will just use 29.
So, basically, 29 is our standard deviation.
Again, as I say, it looks kind of intimidating
at the beginning, but it's not.
As long as you follow the step by step process,
it is actually quite simple.
So the first thing, again,
is to create that ordered set
and put all your ordered numbers here
from smallest to largest,
and then you create the second column
where you are deducting your X value,
basically your mean from your X value.
And then what you want to do
is calculate all the negative numbers,
sum them up,
sum up all your positive numbers and look at the difference.
If the difference is zero, or very close to zero,
you know you have done the right thing.
And again, please keep in mind this will only work
if you have the ordered set done properly.
Okay? If you have not done the order set properly,
it's going to be very difficult
to sum up the negatives on the positives,
not that you cannot do it,
but it'll be much more challenging.
So again, strongly recommend, please do an ordered set.
And then, the third column is basically
doing the whole square.
And as you can see, I did all of those.
Again, please feel free to let me know
if you have any questions,
I'm here to help, as I've said many times,
and I really want you to understand the process.
Okay?
I know we can use statistical software packages,
and they do calculate these things for us,
but at the same time,
we do need to understand the process ourself, how it works,
so if the computer or the statistical software package
gives us a weird answer, we will be like,
"This is not feasible."
So we do need to understand the process
a little bit ourself, how it works.
So, that's about it for 1B,
and now we are going to move to 1C.