ck12.org Normal Distribution Problems: Empirical Rule

Using the empirical rule (or 68-95-99.7 rule) to estimate probabilities for normal distributions

Let's do another problem from
the normal distribution

section of ck12.org's
AP statistics book.

And I'm using theirs because
it's Open Source and it's

actually quite a good book.

The problems are, I think,
good practice for us.

So let's see, number 3.

You could go to their
site and I think you

can download the book.

Assume that the mean eight of 1
year old girls in the U.S. is a

normally distributed-- or is
normally distributed with the

mean of about 9.5 grams.

That's got to be kilograms.

I have a 10 month old son
and he weighs about 20

pounds which is about 9
kilograms not 9.5 grams.

9.5 grams is nothing.

This would be talking about
like mice or something.

This has got to be kilograms.

But anyway, it's about
9.5 kilograms with a

standard deviation of
approximately 1.1 grams.

So the mean is equal to 9.5
kilograms I'm assuming and

the standard deviation
is equal to 1.1 grams.

Without using a calculator-- so
that's an interesting clue--

estimate the percentage of 1
year old girls in the U.S. that

meet the following conditions.

So when they say that without a
calculator estimate that's a

big clue or a big giveaway that
we're supposed to use

the empirical rule.

Empirical rule sometimes
called the 68-95-99.7 rule.

And if you remember this
is the name of the rule

you've essentially
remembered the rule.

What that tells us that if we
have a normal distribution--

I'll do a bit of a review
here before we jump

into this problem.

If we have a normal
distribution-- let me draw

a normal distribution.

It looks like that.

That's my normal distribution.

I didn't draw it perfectly
but you get the idea.

It should be symmetrical.

This is our mean right there.

That's our mean.

If we go one standard deviation
above the mean and one standard

deviation below the mean,
so this is our mean plus

one standard deviation.

This is our mean minus
one standard deviation.

The probability of finding a
result if we're dealing with a

perfect normal distribution
that's between one standard

deviation below the mean and
one standard deviation above

the mean-- that would be this
area-- and it would be,

you could guess, 68%.

68% chance you're going to get
something within one standard

deviation of the mean.

Either a standard deviation
below or above or

anywhere in between.

Now, if we're talking about two
standard deviations around the

mean-- so if we go down another
standard deviation, we go down

another standard deviation in
that direction and another

standard deviation above the
mean-- and we were to ask

ourselves what's the
probability of finding

something within those two or
within that range, then it's,

you could guess it, 95%.

And that includes this
middle area right here.

So the 68% is a
subset of that 95%.

And I think you know
where this is going.

If we go three standard
deviations below the mean and

above the mean, the empirical
rule or the 68-95-99.7 rule

tells us that there is a 99.7%
chance of finding a result in a

normal distribution that is
within three standard

deviations of the mean.

So above three standard
deviations below the mean

and below three standard
deviation above the mean.

That's what the empirical
rule tells us.

Now let's see if we can
apply it to this problem.

So they gave us the mean and
the standard deviation.

Let me draw that out.

Let me draw my axis
first as best as I can.

That's my axis.

Let me draw my bell curve.

That's about as good as a
bell curve as you can expect

a freehand drawer to do.

And the mean here is 9.-- and
this should be symmetric.

This height should be the
same as that height there.

I think you get the idea.

I'm not a computer.

9.5 is the mean.

I won't write the units.

It's all in kilograms.

One standard deviation above
the mean we should add 1.1 to

that because they told us the
standard deviation is 1.1.

That's going to be 10.6.

If we go-- let me just draw a
little dotted line there-- 1

standard deviation below the
mean we're going it subtract

1.1 from 9.5 and so
that would be 8.4.

If we go two standard
deviations above the mean

we would add another
standard deviation here.

Right?

We went one standard
deviations, two

standard deviations.

That would get us to 11.7.

And if we were to go three
standard deviations

we'd add 1.1 again.

That would get us to 12.8.

Doing it on the other side,
one standard deviation

below the mean is 8.4.

Two standard deviations below
the mean-- subtract 1.1

again-- would be 7.3.

And then three standard
deviations below the mean--

which we'd write there--
would be 6.2 kilograms.

So that's our set up
for the problem.

So what's the probability that
we would find a one year old

girl in the U.S. that weighs
less than 8.4 kilograms.

Or maybe I should say
whose mass is less

than 8.4 kilograms.

So if we look here, the
probability of finding a baby

or a female baby who is one
year old with a mass or a

weight of less than 8.4
kilograms, that's this

area right here.

I said mass because kilograms
is actually a unit of mass.

Most people use it
as weight as well.

So that's that
area right there.

So how can we figure out
that area under this

normal distribution using
the empirical rule?

Well, we know what
this area is.

We know what this area between
minus one standard deviation

and plus one standard
deviation is.

We know that is 68%.

And if that's 68% then that
means in the parts that

aren't in that middle
region you have 32%.

Because the area under the
entire normal distribution is

100 or 100% or 1, depending on
how you want to think about it.

Because you can't have-- well,
all of the possibilities

combined can only add up to 1.

You can't have it more
than 100% there.

So if you add up this leg and
this leg-- so this plus that

leg is going to be
the remainder.

So 100 minus 68, that's 32.

32%.

32% is if you add up
this left leg and this

right leg over here.

And this is a perfect
normal distribution.

They told us it's
normally distributed.

So it's going to be
perfectly symmetrical.

So if this side and that side
add up to 32 but they're both

symmetrical, meaning they have
the exact same area, then this

side right here-- I'll do it in
pink-- this side right here--

it ended up looking more
like purple-- would be 16%.

And this side right
here would be 16%.

So your probability of getting
a result more than one standard

deviation above the mean-- so
that's this right hand

side, would be 16%.

Or the probability of having a
result less than one standard

deviation below that mean,
that's this right here, 16%.

So they want to know the
probability of having a

baby at one years old
less than 8.4 kilograms.

Less than 8.4 kilograms
is this area right here.

And that's 16%.

So that's 16% for part a.

Let's do part b: between 7.3
and 11.7 point seven kilograms.

So between 7.3--
that's right there.

That's two standard deviations
below the mean-- and 11.7, one,

two standard deviations
above the mean.

So there's essentially asking
us what's the probability of

getting a result within two
standard deviations

of the mean, right?

This is the mean right here.

This is two standard
deviations below.

This is two standard
deviations above.

Well that's pretty
straightforward.

The empirical rule tells us
between two standard deviations

you have a 95% chance of
getting a result that is within

two standard deviations.

So the empirical rule just
gives us that answer.

And then finally, part c: the
probability of having a one

year old U.S. a baby girl
more than 12.8 kilograms.

So 12.8 kilograms is three
standard deviations

above the mean.

So we want to know the
probability of having a result

more than three deviations
above the mean.

So that is this area way out
there that I drew in orange.

Maybe I should do it in
a different color to

really contrast it.

So it's this long tail out
here, this little small area.

So what is that probability?

So let's turn back to
our empirical rule.

Well we know the probability--
we know this area.

We know the area between minus
three standard deviations and

plus three standard deviations.

We know this-- since this is
last the last problem I can

color the whole thing in-- we
know this area right here

between minus 3 and plus
3, that is it 99.7%.

The bulk of the results
fall under there.

I mean, almost all of them.

So what do we have left
over for the two tails?

Remember there are two tails.

This is one of them.

Then you have the results that
are less than three standard

deviations below the mean.

This tail right there.

So that tells us that this,
less than three standard

deviations below the mean and
more than three standard

deviations above the mean
combined have to be the rest.

Well the rest, there's only
0.3% percent for the rest.

And these two things
are symmetrical.

They're going to be equal.

So this right here has to be
half of this or 0.15% and

this right here is
going to be 0.15%.

So the probability of having a
one year old baby girl in the

U.S. that is more than 12.8
kilograms if you assume a

perfectly normal distribution
is the area under this curve,

the area that is more than
three standard deviations

above the mean.

And that is 0.15%.

Anyway, I hope you
found that useful.