Welcome to Highline BI-348, video number 24.
Hey, if you want to download this workbook-- BI-348 chapter
two start, or the finished file, click on the link
below the video.
In this video we want to talk about the Empirical Rule.
Now, back in our prerequisite class,
we spent chapters upon chapters using the empirical rule.
And it's a great model-- and here's the model--
because we can calculate probabilities easily using
the Empirical Rule and our normal curve.
However, we have to make sure that the distribution has
a symmetrical, or near symmetrical, distribution.
So often times people use histograms.
We also talked in the prerequisite class
about the Central Limit Theorem that
allowed us, for sample data, to use the normal distribution.
Now in this video, we want to practice
calculating the probabilities.
I actually have one, two, three yellow sheets here--
they actually should be blue.
And I want to click on each one and then just go over our three
possibilities for calculating probability
using the Empirical Rule.
Now here's our bell-shaped distribution.
And we have past test scores and the distribution was normally
distributed, or symmetrical.
We know what the mean is.
So the mean for past scores was 74,
the standard deviation is 10, and we
need to figure out what the probability of getting
a score of 64 or less is.
Now, actually, this is the first of three possibilities.
We can actually calculate probability at a particular
x or less.
We can calculate a particular-- this case
we could take the same x-- we could calculate
the probability of a particular x or more,
or we can do between two values.
Remember with a continuous distribution like this
we can't calculate an exact x.
We're always calculating between two points.
Or, in this case, between an x all the way down
to negative infinity.
Now we use our NORM.DIST or NORM.INV.DIST We give it values
and it gives its probability.
NORM.S.INV inverse-- we give it probability
and it gives us a value.
The s functions, .S.DIST, inverse.
S.INV, those are when we have the z values.
So we're going to do this example both ways.
Once with just the distribution function
and once with the S.DIST.
So we're going to select NORM.DIST.
The screen tip is pretty polite.
It wants the particular x.
Here's our particular x.
Comma, the mean, there's the mean.
Comma, standard deviation-- boom,
there's the standard deviation.
And then we have a choice.
So false, or zero, is the probability mass function,
and that is only for charting.
We are going to be using the cumulative distribution
function, and I'm always going to put, instead of true, a one.
Now before I hit Enter, we have to remind ourselves
the NORM.DIST, and actually all the distribution functions
in Excel always calculate from the lowest value
up to the x value we give it.
So notice, we gave it a 64, so it's
going to calculate from negative infinity--
because this hypothetically extends forever--
all the way from negative infinity up to that 64.
So when we hit Enter, that's the probability from 64 or below.
About 15.9%.
Now we can calculate our z.
Remember, that is we take the particular value
minus the mean, that right there gives us the deviation,
so if I entered this, that's minus 10.
F2, and we want it in terms of standard deviation,
so I divide it by standard deviation.
And this one's going to be easy.
We could do this one in our head, right?
-1.
64 is minus 10 below 74, so, of course, when you divide it
by a standard deviation of 10, it
gives us exactly one standard deviation below the mean.
Now, if we're going to use our NORM.S functions,
that means the input is going to be a z value.
We give it the z, that's all we need,
because remember that value already
has in it the mean, the particular value,
and the standard deviation.
Comma, same thing we're using one
to calculate the probability.
And it gives us exactly the same answer.
Now let's go to our next example.
Here's, same test scores, 74 is the mean.
Standard deviation is 10.
And this person got 80.
And so we want to calculate the probability
of getting an 80 or above.
Equals NORM, and we're going to use our DIST.
Now we have to be careful, because the functions always
go from negative infinity up to the x we give it.
But let's go ahead and-- hey there's the x.
That's the 80.
Comma, the mean, comma, the standard deviation, comma
and one for cumulative distribution.
Now if I throw an 80 in here, it's
going to give me the probability from negative infinity
all the way up to the 80.
That's not what we want.
We want everything on the upper end.
So of course, we use the complement rule,
and take one, which represents 100 percent,
and subtract probability from negative infinity
up to that 80, when I Control-Enter,
that's the probability of getting a score of 80 or above.
Now we can calculate a z, particular value
minus the mean, close parentheses.
And we want it in terms of standard deviation,
so we divide it by the standard deviation.
0.6 standard deviations above the mean.
Now we can calculate the probability
of getting 0.6 z-score or more.
Equals-- and we're on the upper end-- so one minus NORM.S.DIST.
There's our z, comma, 1 close parentheses, Control-Enter.
We get exactly the same answer.
Now we want to go over to this sheet normal between.
Calculating probability between.
We have the same distribution.
74 is the mean.
Standard deviation is 10.
And we want to calculate the probability of getting
a score between 90 and 75.
Well, if the functions always go from negative infinity
up to some x, then the way we do between
is we go from negative infinity up to the bigger, that's
all the area below, and we subtract
a second normal distribution function
that'll go from negative infinity up to the lower.
So imagine all of this subtract all of that
gives us the difference.
So equals NORM.DIST, the x is going to be the bigger.
Comma, mean, comma, standard deviation,
comma, 1 for cumulative.
And we subtract NORM.DIST.
The lower, that's the smaller x, comma, mean, comma,
standard deviation, comma, 1, close parentheses.
That will give us the difference.
So the probability of getting a score between 90 and 75
is about 40%.
Now we can calculate two Zs also.
Equals the deviation, and this one's
the lower particular value minus the mean.
It looks like we're exactly one point above the mean.
We divide that by standard deviation.
And there we go.
0.1.
That's number of standard deviations.
Z equals the particular value, the bigger one,
minus close parentheses, divided by our standard deviation.
And it looks like, with a score of 90,
you're 1.6 standard deviations above the mean.
Now we can calculate the difference.
And it still means two functions, but since it's an S,
we're putting in Zs.
It's always the bigger one, comma one, close parentheses.
And we add a second NORM.S.-- for z- .DIST.
And the z, that's the smaller one, comma one,
close parentheses.
And guess what.
We get exactly the same answer.
About 40% probability, your chances
of getting the score between 75 and 90.
Now let's go over to the sheet Empirical Rule practice.
I just looked up Seattle's mean gas price today
and it's $2.67.7.
And the standard deviation's pretty big,
because Seattle has a lot of different neighborhoods
with different gas prices.
So we want to first try and to calculate the probability
of finding a gas station, randomly selecting one,
that has a gas price of $2.20 or less.
So equals, and we use our NORM.DIST.
There's our x, comma, there's our mean, comma, standard
deviation, comma, one.
And this is below, so we don't have to do anything, just
Control-Enter.
Wow.
So the probability of randomly selecting a gas station
in Seattle and finding $2.20 or less is like 2.6%.
Now what's the z?
Equals a particular value minus the mean,
and we want to divide it by standard deviation.
So that's-- wow-- that's almost two standard deviations below
the mean.
With that z, we can use the NORM.S.DIST.
There's the z, comma one, and Control-Enter.
We get exactly the same thing.
How about randomly selecting a gas station and finding that it
is selling gas for $2.90 or above?
Equals NORM.DIST-- I'm already getting in trouble,
because guess what, I forgot my -1, but no problem.
That function will give me everything from negative
infinity up to $2.90.
I use the complement rule and boom.
So about 10.7% chance that I could randomly select the gas
station find $2.90 or more.
And the z-score, number of standard deviations.
We're going to take the particular value
and subtract the mean and divide it by the standard deviation.
So this one is 1.24 standard deviations above.
Equals one minus NORM.S. There's a z, comma one, boom.
We get exactly the same thing.
Now here's two other prices.
Let's calculate the z for each one of these.
$2.18.7, so equals particular value, one cell above,
minus the gas price locked with the F4 key, close parentheses,
divided by the standard deviation
locked with the F4 key.
Control-Enter, Control-C, Arrow, Arrow, Control-V. Now z
of exactly -2.
So this $2.18.7 is exactly two standard deviations
below the mean.
And this 3.067 is exactly two standard deviations
above the mean.
We don't even have to do the functions to estimate,
because our Empirical Rule says 68% of the values will
lie between plus or minus one standard deviation.
About 95% of the data values will
lie within plus or minus two standard deviations.
And here's our rule 99%, or practically all values,
will lie between plus or minus three standard deviations.
So if I come back over here, my estimate is going to be 0.95.
I'll put my estimate down here.
But, of course, we can use the NORM.S.DIST
and get the bigger z, comma one, close parentheses.
And then second NORM.DIST, and get our smaller z, comma one,
close parentheses, Control-Enter.
And there is a more precise measure.
If we use our NORM functions, or any of the distribution
functions, as opposed to using a rule or tables in a book,
we're going to get a more accurate probability.
All right.
In this video, we reviewed our awesome normal distribution
functions for calculating probability
when our distributions have a normal distribution.
And we talked about the Empirical Rule.
All right.
We have two videos left in this chapter,
and when we come back, in our next video,
we'll talk about percentiles and quartiles.
All right we'll see you next video.