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So next we're going to look at a different measure of, of spread called
the standard deviation. and so we're going to be measuring
deviation or difference from the mean. And so the first thing we're going to
need here is the mean of these scores. and the mean of these scores turns out to
be, turns out to be five. And so what we're going to do here is
look at our data, and then the deviation, in other words the difference between the
data and the mean. How far each of my data values is from
the mean. so for this data set here.
Let's see we got zero, zero, zero, zero, zero, ten, ten, ten, ten, ten so the
deviation here is data minus mean is negative five.
And I think you can trust me that all of these other zeroes are going to be the
same. and then for the 10, 10 minus 5 is, is,
is 10 minus 5 is also This five, positive five, and so one down the line.
Now we want to find some measure similar to sort of the, the average of our
deviations. The problem is that if we added these all
up, they'd add up to zero. and so we need to do something else.
And there's a couple of approaches that we could use.
It turns out the one that makes the most sense is to To square the deviations.
There's various reasons for this, but most simply it, it will definitely make
the values positive. So.
Negative 5 squared is 25. 25, 25, 25, 25, 25, 25, 25, and in this
particular case it's very easy to calculate too.
Uh,[LAUGH] and so we find all of our deviations there.
So now we're going to find the sum of those squared deviations.
So we're going to add all these up. So we're going to add all these up let's
here, I'm adding up 25 10 times, right? So that ends up being 250.
So we're going to find the sum of the squared deviations Sure I'm going to
right that down here, 250. And them I'm going to divide it by not
how many not by how many doll, okay. Well what we divide by depends upon
whether our data was coming from a sample, in other words from a sampling of
all of the data. or whether it was representing the whole
population. So, if it's the population, then we
divide by N the number of data. If it's the sample, then we divide by By
n minus 1. 1 less than the total number of data.
in this case, this is representing the entire section's, scores.
And so we're going to divide by n. We're going to divide by, the.
We're going to divide by n, we're going to divide by the 10 scores and
we're going to get 25. Now at this point we have something
called variance. or more specifically the population
variance since, since this was for the entire population.
The standard deviation, standard deviation, deviation is the square root.
Of the variance, so it's the square root of 25.
The square root of 25 is 5. And so the standard deviation of this
data set is 5 5 units. Now for comparison, let's look at all of
our other data scores. When all of our scores are the same The
devialtion is going to be 0. The data set that we just looked at had a
deviation of 5. In this case all the scores are much
closer to the mean and so the standard deviation is going to be smaller.
In this case even though we have the same mean and median.
and range as our data set B, the standard deviation here is much smaller because
most of the data is very close to the mean and we only have a few values that
are further out. So if the data's very spread out we have
a big standard deviation. If the data is closer together we have a
small standard deviation.