>> So in this video I'm going to show you how to draw the normal.
Okay, so what I'm going to do is I'm just going
to show you how you would draw the normal
and also some thoughts behind drawing the normal.
So usually in the word problem you find a
particular probability.
Let's say you had the probability of x being greater
than or equal to let's say 20 and let's say
that the population mean let's say that was 30
and the population standard deviation let's say was 10.
These pieces of information are very important
in creating a normal.
So I'm going to draw a line and I'm going
to name it the merandum [phonetic] variable x okay.
And this might not be sufficient, but we'll see
if I'm able to have enough space for my number line.
First thing I want to do is to draw 30,
which would be let's say here.
Thirty is my mean and that's
where my curve will be drawn, right.
Well that's horrible, but that's the idea.
There's some point in here right, some point, the top part
of the normal and somehow it bends right down.
There's some other point as well,
an inflection point, right.
There's a vertex right the top point, but then there's the,
there are these inflection points meaning
that the curve is going you know, going down perfectly
and shifts and curves right, that pitted point.
Where are these points of inflections?
Well first of all where's this you know particular point,
the top portion of the normal?
Well that one's easy.
You know where that is because its right at the top
of the true mean, which is right here, we'll say it's over here.
Okay now again we have this curve right
and there's some inflection point.
Where are they?
Well turns out that these inflection points are
about one standard deviation from the left and one
from the left and from the right.
So, negative one standard deviation from the left
and one standard deviation to the right.
That's where your inflection point is okay
and that would be your value.
Your first standard deviation,
one standard deviation away from the mean.
So if that's the case, then this would be 40
because the standard deviation is 10,
so you would add 10 to 30, right.
And of course that would give you 40.
And then you would be two standard deviations away, right.
And that would give you, you would add another 10 to 40
or you would add 20 to 30
because 30 plus one standard deviation plus another standard
deviation and then that would be 50, right.
And then usually on a normal you have three standard deviations,
right.
That would be 30 plus another 10 plus 10 plus 10.
One standard deviation, two standard deviations,
three standard deviations.
So that would be 30, 40, 50, 60.
So 60 would be over here.
That's really messy, that's the idea
and of course you would also do it the other way.
So let me erase this here and.
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So I'm going to draw a dot
at one standard deviation, so how far is that?
Well in this graph we'll say
that every two spaces here is 10.
So this would be 40, so this would be 40, two more spaces,
50, two more spaces 60 and then the other way.
Two spaces down, right will give you 20.
Two more spaces down will give you 10.
Two more spaces down will give you zero, perfect.
So again we have this curve right.
Somewhere it bends right because of an inflection point.
Where is that point?
Well it's one standard deviation away from the mean
and how far away is one standard deviation
to the right and to the left?
That's right, 10 away, 10 away from 30.
So you would draw a dot somewhere around here
and then another dot over here, asymmetrical.
So you can see that its one standard deviation
to the left and to the right.
A curve right.
Maybe I can draw a little bit better.
Okay that's pretty close.
That's the top portion of your normal and then what do we do?
Well then we need to do the tails right.
So this piece right here is this way,
now it needs to go that way right.
So how do we do that?
Well we draw it and what we're doing is we're aiming here while
we're going down.
It's about right.
Now this line at three standard deviation is getting very very
close to the x axis.
It never crosses or it never touches it,
so make sure you don't do that.
And then we're going to draw it on this side as best we can.
That's pretty symmetrical.
Not perfect, but pretty good.
And what you can do is you can erase you know, these dots.
It's not a perfect drawing.
Some people can draw it very well.
I can't, but that's not bad and if you wanted to keep going
with this right, what's the probability
of x being greater or equal to 20?
You know it would be here would be the line
and then you would shade it.
Now I have another video I'm going
to show you how we do that.
The probability of let's say z is less than or equal to 1.84.
Now here the random variable is z,
so we now that there are z values, z scores,
z table, standard normal.
And what is the standard normal's population mean
standard deviation?
Give you a minute, less than a minute.
That's right, population mean is going to be zero
and the standard deviation is going to be one.
Okay so just because I knew that the random variable was z
or z values, z scores, what I'm going
to do this is a standard normal curve.
They're all written the same way.
Okay, so here's my line.
You can draw a little bit better.
And at the middle right,
right in the center is going to be zero.
That's my population mean, right.
Question is how far, how many numbers do I write
to the right or to the left?
Most normal's right, only go up to three standard deviations.
So in this case I'm going to write one, two,
three and I spaced it out using two blocks.
Same thing over here, this is going to be negative one,
negative two and negative three okay.
Now it's the same idea.
The question is where are no,
it's going to be something like that.
How can we draw it a little bit better?
Well first of all we know we're going to need a dot right there,
right above the population mean and then we want
to be one standard deviation away right from the mean.
This is the mean, we want to be one standard deviation
to the right, one standard deviation to the left.
Now a question might be well where do you draw these dots?
Do you draw it in here, you draw them here?
Well you get a sense for example if you draw them there,
it's going to be kind of a little bit harder to draw maybe.
If you draw, if you put them too low, then it's going
to look kind of funny right.
Engage it, okay at this particular time you're beginning
understand how to draw these things and you gauge the dots.
And then we draw the top part right.
This is some type of vertex right, top part
and then we draw a last piece.
Again that's this right and now we're drawing a bottom part.
We saw it changes direction and well that's pretty close.
Again we don't want to cross the line right.
It's very close to the line.
Same thing over here and draw it there.
Maybe we don't know, again you try to do your best.
And then you can erase these over here.
And as you probably already know right, the empirical rule,
when you're one standard deviation away,
let's try a different color here.
We'll draw it with the orange.
When you're one standard deviation away
on the mean right, one standard deviation from the right,
then one standard deviation to the left right.
This area right here, that's going to be 68%, right.
And when you are two standard deviations away,
two standard deviations away you're about right,
about it's a little bit more, but about 95% okay.
This is not, this is not the correct value,
but it's an estimate and it's really 1.96
or something like this.
That's where at 95% on both sides made a 1.96, but you know
for an estimate remember this is a stats score right
we're estimating.
We're trying kind of like keep these numbers in our head okay.
Course you'll take an exam you have to be more accurate
and then when you're three standard deviations away,
when you shade it, right all this shade this is
about oh what 99% probably okay.
Now we're going to talk about shading the normal.
Now one thing that I missed here right,
this is the random variable of z right.
Z for the z score right or the random variable z,
so that you can look it up in your z table.
So this gives you kind of like an understanding on how
to draw a normal, right a regular normal and also how
to draw a standard normal using two values right.
Population mean and standard deviation
and as you know you only need two pieces
of information to draw the curve.
And what I just was demonstrating gives you an idea,
a sense of how to draw these.