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## Normal distributions and the empirical rule

# Normal distribution problems: Empirical rule

AP.STATS:

VAR‑2 (EU)

, VAR‑2.A (LO)

, VAR‑2.A.3 (EK)

CCSS.Math: ## Video transcript

Let's do another problem from
the normal distribution section of ck12.org's AP
Statistics book. And I'm using this
because it's open source. It's actually quite a good book. The problems are, I think,
good practice for us. So let's see, number
three, number two. You can go to their
site, and I think you can download the book. Assume that the mean weight of
one-year-old girls in the US is normally distributed with
a 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. 9.5 grams is nothing. This would be if we were 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 one-year-old
girls in the US that meet the following condition. 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, 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 is 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. So 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. So 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 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
deviations 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. Let me draw the bell curve. That's about as
good of a bell curve as you can expect a
freehand drawer to do. And the mean here is-- 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. Let me just draw a
little dotted line there. Once standard deviation
below the mean, we're going to
subtract 1.1 from 9.5. And so that would be, what? 8.4. If we go two standard
deviations above the mean, we would add another
standard deviation here. We went one standard deviation,
two standard deviations. That one goes 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 that 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, it would be right there,
would be 6.2 kilograms. So that's our setup
for the problem. So what's the
probability that we would find a one-year-old
girl in the US 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 that's
one-years-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. But most people use
it as weight, as well. So that's in 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 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 the possibilities combined can only add up to 1. You can't have 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%. And 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-- 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 the 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
kilograms-- so between 7.3, that's right there. That's two standard
deviations below the mean. And 11.7-- it's two standard
deviations above the mean. So they're essentially
asking us what's the probability of getting
a result within two standard deviations of the mean. This was 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 bad results,
or 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 US 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 standard
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 this area. We know the area between minus
three standard deviations and plus three
standard deviations. We know this. Since this is the last problem,
I can color the whole thing in. We know this area, right here--
between minus 3 and plus 3. That is 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. And 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--
it's only 0.3%. for the rest. And these two things
are symmetrical. They're going to be equal. So this right here it 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 US that is
more than 12.8 kilograms, if you assume a perfect
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, hope you
found that useful.