hello and welcome to my secret math tutor in this video we're gonna cover
how to use the empirical rule for normal distributions. Now first we should talk
just a little bit about what this actually says and then we'll get into a
couple of examples on how to use it. So the empirical rule just gives us how
much of our data lies within one two or three standard deviations of the mean.
So literally if you want to take a look at this you know in the center of each
of these normal distributions I have the mean and then we're looking at one
standard deviation of either side two standard deviations of either side or
three standard deviations of either side and what the empirical rule says is that
we could expect 68% of our data to be within one standard deviation
95% of our data to be within two standard deviations and
99.7% to be within three standard deviations so it's
just really just a wonderful rule to tell us how much of our data we can
expect to be around our mean all right let's get into some examples and see how
this might work for the first example we're gonna look at grades for a certain
history exam and we're given that they follow a normal distribution that's
important because this empirical rule works for normal distributions looks
like we have a mean of 78 and a standard deviation of 6 so our goal is to find
the range of values around the mean that include 95% of the grades so just as a
quick things to get started it's a good idea to identify key bits of information
like the mean and the standard deviations well let's go ahead and just
write those down so our mean is 78 under standard deviation is 6 now we're
looking for the range around the mean which includes 95% of the grades if we
were called back to what I just said about the the empirical rule 95%
around the mean that corresponds to two standard deviations on either side of
our mean so what we really need to do is just take our mean and if we're gonna
look for that range will either add or subtract two standard deviations around
it all right let's get that bottom range first so our mean is 78 we will subtract
two standard deviations or two times 6 so I'll be 70 minus
twelve so our lower number is 66 to get the upper number almost identical 78
now we're adding two standard deviations or plus two times six so that's 78 plus
twelve and I'll give us 90 alright so putting it all together this tells us
that grades between 66 and 90 that if I have grades between 66 and 90 that will
capture 95% of all the grades now this is just an approximate value know if you
were looking at real data it might not be exactly 95% but it is probably going
to be pretty close all right let's look at one more example in this example
we're looking at the heights of women and it looks like they follow a
bell-shaped distribution remember that's just another way of saying that this is
a normal distribution looks like it has a mean of 160 centimeters and the
standard deviation of seven point five it looks like we want to know what is
the approximate percentage of women between 137.5 centimeters and 182 point
five centimeters all right so like before again always a good idea identify
what the mean and standard deviation is when you're given that information
especially for bell shape or normal distributions so we have a mean of 160
centimeters standard deviation of seven point five centimeters all right and
we're given a range and we're kind of backtracking trying to figure out what
that percentage is well since this is a bell-shaped distribution normal
distribution we're still going to think of the mean and either add or subtract
some number of standard deviations and connect that into our range all right so
let's see let's go ahead and borrow the upper range because that'll allow us to
figure out how many standard deviations we are away so that's the mean 160 we
don't know how many standard deviations are we are so we'll just leave that as K
and we'll multiply that by the standard deviation of seven point five and since
we're adding this should equal the upper number
the one 82.5 all right now we just need to solve this for K and see how many
standard deviations that is we'll start that by subtracting the 160 from both
sides no problem that'll give us twenty two
point five and then we can divide both sides of this by seven point five that
will give us that K is equal to three perfect so there's three standard
deviations to reach that upper range alright so thinking back to the
empirical rule here's where we can use it ninety-nine point seven percent that
corresponds to three standard deviations nice so what does this really tell us
what was the approximate percentage of women between these two values well
according to the empirical rule its ninety-nine point seven percent and
again that might not be exact but it is a really good approximation all right
hopefully that helped you out understand more about the empirical rule if you'd
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