in this video we'll take a look at how to calculate the standard deviation and
the variance the standard deviation and variance are measures of variability
they both indicate the overall amount of variability or spread in a set of scores
the two measures are closely related the standard deviation is just the square
root of the variance the standard deviation can be thought of
as the average or typical distance that values vary from the mean a standard
deviation of two for example would indicate that scores vary around the
mean by about two points so if we had a standard deviation of two that would
mean that some values were further than two points away from the mean and some
values were closer than two points but loosely speaking the average distance
was about two points from the mean so some are a little more some are little
less some may be exactly two points away but sort of an average distance is about
two points that's what we're getting out here with the standard deviation next
we'll take a look at how to calculate the standard deviation and variance in
this example we'll look at the population variance and standard
deviation we'll calculate the sample values in another video
so suppose that we have the following values with scores for five people on
the variable X we'll go ahead and calculate the variance and standard
deviation on this example in terms of the calculations we'll begin with
calculating the mean of X so to find the mean of X we want to go ahead and add
all these values together and then divide by five the total number of
values so you can see we did that here we have ten 18 and 20 divided by five
which gives us a mean of four next we need to calculate deviation scores
deviation scores subtract the mean from each value
they're called deviation scores because they indicate how far each value
deviates or departs from the mean so in other words how far is each value away
from the mean that's what a deviation score measures let's go ahead and take a
look at how to find those okay so here I have my deviation scores
and recall that my mean is 4 so here's the original values and my mean is 4 so
what I'm going to do is I'm going to subtract this mean of 4 from each of
these values and you can see that here so I have 6 minus 4 is 2 4 minus 4 is 0
5 minus 4 is 1 3 minus 4 is a negative 1 and then 2
minus 4 is a negative 2 now if you look at these deviation scores notice we have
a plus 2 0 plus 1 negative 1 negative 2 notice how the positive deviation scores
add up to 3 and the negative deviation scores add up to negative 3 if you add
up all of these they sum to 0 now that's always the case with deviation scores
and I summarized that here when we add these values up positive 3 negative 3
they always equal 0 and the reason that this is always the case is because the
mean acts as a balance point in the distribution with the sum of the
deviation scores above the mean equal to the sum of the deviation scores below
the mean that's always the case if you calculate deviation scores and find that
they don't add up to 0 then that means that a mistake was made in the
calculations and you want to double check your work so they should always
add to 0 by definition okay after we find the deviation scores
we need to square each of these values and squaring the deviation scores gets
rid of the negative numbers so let's take a look at that these first
two columns you saw in the earlier slide where we calculated the deviation scores
here now all we're doing is we're taking these deviation scores and we're
squaring them and that's what we have right here 2 squared is 4 0 of square
root of 0 1 squared is 1 negative 1 squared is positive 1 and negative 2
squared is positive 4 so notice how we have all positive values now
next we need to sum the squared deviation scores this is also known as
the sum of the squares or SS and you'll see this term a lot in Statistics texts
SS for sum of the squares and here's our last column in our previous calculations
where we squared each of those deviation scores all we need to do is add these
values together so four plus zero plus one plus one plus four and we've done
that right here this gives us the sum of the squares and when we add these
together 5 6 + 4 is 10 so notice that the sum of the squares or SS is 10 with
SS solved for we can now calculate the variance the formula for the variance is
the variance is equal to SS over N and here's our values that we saw before SS
is 10 n is 5 as a reminder in just counts how many values we have so
there's five values so to calculate the variance we're just going to take 10 and
divide it by 5 and that gives us a variance of two
the standard deviation is just the square root of the variance so this is
the standard deviation here and it's equal to the square root of the variance
recall that the variance was 2 so therefore the standard deviation is
equal to the square root of 2 which works out to be 1 point 4 1 4
in summary we had a variance of two and a standard deviation of one point four
one four now as a brief review before we close to
calculate the standard deviation of variance you want to complete the
following steps we found the mean and then recall we calculate a deviation
scores where we subtracted the mean from each of these values and then we squared
those deviation scores and then we added up those squared values and that gave us
SS and then we just divided SS by n to find the variance and then once we found
the variance we took the square root of that value and that gave us the standard
deviation okay that's it this concludes the video on calculating the standard
deviation and variance thanks for watching