ck12.org Exercise: Standard Normal Distribution and the Empirical Rule

We're now on problem number 4
from the Normal Distribution

chapter from ck12.org's
FlexBook on AP Statistics.

You can go to their
site to download it.

It's all for free.

So problem number 4, and
it's, at least in my mind,

pretty good practice.

For a normal, or a standard
normal distribution,

place the following in order
from smallest to largest.

So let's see, percentage of
data below 1, negative 1.

OK.

Let's draw our standard
normal distribution.

So a standard
normal distribution

is one where the
mean is-- sorry,

I drew the standard
deviation-- is

one where the mean, mu for
mean, is where the mean is

equal to 0, and the standard
deviation is equal to 1.

So let me draw that standard
normal distribution.

Let's see, so let me draw
the axis right like that.

Let me see if I can draw
a nice-looking bell curve.

So there's the bell
curve right there.

You get the idea.

And this is a standard normal
distribution, so the mean,

or you can kind view the
center point right here.

It's not skewed.

This says the mean is
going to be 0 right there,

and the standard deviation is 1.

So if we go 1 standard
deviation to the right,

that is going to be 1.

If you go 2 standard
deviations, it's

going to be 2, 3 standard
deviations, 3, just like that.

1 standard deviation to the
left is going to be minus 1.

2 standard deviations
to the left

will be minus 2, and
so on, and so forth.

Minus 3 will be 3 standard
deviations to the left

because the standard
deviation is 1.

So let's see if we can
answer this question.

So what's the percentage
of data below 1?

Part a, that's this
stuff right here.

So everything below 1, so
it's all of-- well, not

just that little center portion.

It keeps going.

Everything below 1,
percentage of data below 1.

So this is another
situation where

we should use the
empirical rule.

Never hurts to
get more practice.

Empirical rule, or
maybe the better way

to remember the empirical rule
is just the 68, 95, 99.7 rule.

And I call that a
better way because it

essentially gives you the rule.

These are just the
numbers that you

have to essentially memorize.

And if you have a calculator
or a normal distribution table,

you don't have to do this.

But sometimes in
class, or people

want you to estimate
percentages,

and so you can impress people
if you can do this in your head.

So let's see if we can use
the empirical rule to answer

this question, the area under
the bell curve all the way up

to 1, or everything
to the left of 1.

So the empirical rule tells
us that this middle area

between 1 standard
deviation to the left

and 1 standard deviation to the
right, that right there is 68%.

We saw that in the
previous video as well.

That's what the
empirical rule tells us.

Now, if that's 68%, we
saw in the last video

that everything else
combined, it all

has to add up to 1 or to 100%,
that this left-hand tail-- let

me draw it a little
bit-- this part

right here plus this part
right here has to add up,

when you add it to 68, has
to add up to 1 or to 100%.

So those two combined are 32%.

32 plus 68 is 100.

Now, this is symmetrical.

These two things
are the exact same.

So if they add up to 32,
this right here is 16%,

and this right here is 16%.

Now, the question,
they want us to know

the area of everything--
let me do it

in a new color--
everything less than 1,

the percentage of data
below 1, so everything

to the left of this point.

So it's the 68%.

It's right there,
so it's 68%, which

is this middle area within
1 standard deviation,

plus this left
branch right there.

So 68 plus 16%, which is what?

That's equal to 84%.

So part a is 84%.

They're going to want us to
put this in order eventually,

but it's good to just
solve because that's really

the hard part.

Once we know the numbers,
ordering is pretty easy.

Part b, the percentage
of data below minus 1.

So minus 1 is right there.

So they really just want
us to figure out this area

right here, the percentage
of data below minus 1.

Well, that's going to be 16%.

We just figured that out.

And you could have already
known just without even knowing

the empirical, just looking
at a normal distribution,

that this entire area, that
part b is a subset of part a,

so it's going to be
a smaller number.

So if you just have
to order things,

you could have made
that intuition,

but it's good to do practice
with the empirical rule.

Now part c, they want to
know, what's the mean?

Well, that's the easiest thing.

The mean of a standard
normal distribution,

by definition, is 0.

So number c is 0.

d, the standard deviation.

Well, by definition,
the standard deviation

for the standard normal
distribution is 1.

So this is 1 right here.

This is easier than I
thought it would be.

Part e, the percentage
of data above 2.

So they want the
percentage of data above 2.

So we know from the
68, 95, 99.7 rule

that if we want to
know how much data

is within 2 standard
deviations-- so let

me do it in a new color.

Let me do it in a more
vibrant color, green.

If we're looking from
this point to this point--

so it's within 2 standard
deviations, right,

the standard
deviation here is 1--

if we're looking within
2 standard deviations,

that whole area right there,
by the empirical rule, is 95%,

within 2 standard deviations.

This is 95%.

Which tells us that
everything else combined--

so if you take
this yellow portion

right here and
this yellow portion

right here, so everything
beyond 2 standard deviations

in either direction-- that
has to be the remainder.

So you know everything
in the middle

was 95 within 2
standard deviations.

So that has to be 5%, if you
add that tail and that tail

together, everything to
the left and right of 2

standard deviations.

Well, I've made the
argument before, everything

is symmetrical.

This and this are equal.

So this right here
is 2 and 1/2%,

and this right here
is also 2 and 1/2%.

So they're asking
us the percentage

of data above 2, that's this
tail, just this tail right

here, the percentage of data
more than 2 standard deviations

away from the mean.

So that's 2 and 1/2%.

Let me do it in a darker
color-- 2 and 1/2%.

Now, they're asking
us, let's see, place

the following in order
from smallest to largest.

So there's a little
bit of ambiguity here.

Because if they're saying the
percentage of data below 1,

do they want us to
say, well, it's 84%.

So should we consider
the answer to part a, 84?

Or should we consider-- if
they said the fraction of data

below 1, I would say 0.84.

So it depends on how they
want to interpret it.

Same way here.

The percentage of data below
minus 1, I could say the answer

is 16.

16 is the percentage
below minus 1.

But the actual number, if
I said the fraction of data

below minus 1, I would say 0.16.

So this actually would be very
different in how we order it.

Similarly, if someone
me asked me the percent,

I'd say, oh, that's 2.5.

But the actual number is 0.025.

That's the actual fraction
or the actual decimal.

So I mean, this is
just ordering numbers,

so I shouldn't fixate
on this too much.

But let's just say that
they're going with the decimal.

So if we wanted
to do it that way,

they want to do it from smallest
to largest, the smallest

number we have here is c, right?

That's 0.

And then the next smallest
is e, which is 0.025.

Then the next smallest
is b, which is 0.16.

And then the next one after
that is a, which is 0.84.

And then the largest would
be the standard deviation, d.

So the answer is c, e, b, a , d.

And obviously, the
order would be different

if the answer to this,
instead of saying it's 0.84,

if you said it was 84 because
it's asking for the percentage.

So a little bit of ambiguity.

If we had a question
like this on the exam,

I would clarify that
with the teacher.

But hopefully you
found this useful.