>> In this tutorial we continue our discussion of variability,
also known as dispersion.
Previously, we discussed other measures of variability;
the index of qualitative variation and ranges.
And specifically with ranges, we talked about the range
and the interquartile range.
In this tutorial we'll talk about the final measure,
variance and standard deviation.
Variance and standard deviation are very similar,
they're related.
And were going to talk about them in one video
because you'll see how related they are.
We're going to calculate the variance and standard deviation
for the variable age that was collected on ten students
from statistics class.
So here we have our data, and we're going
to calculate variance with this formula.
Variance equals each value of x minus the mean of x squared,
all of those differences summed and then divided by n minus 1.
And if we look at this formula and break it down for a moment,
we can see that we're adding up something and dividing by n,
in this case n minus 1, the total number minus 1.
Well, we have calculated something similar
to this, a mean.
A mean is a measure of central tendency in which we add
up all the values of x and divide by the total.
This is very similar.
We're calculating a mean of sorts,
but rather than the mean value,
we're calculating the mean differences between each value
and the average, the central measure of central tendency
to mean, the squared differences.
So we're calculating the variance
which is the average squared difference.
So let's see how we do that.
So the formula, again,
is the sum of each x minus the mean squared over n minus 1.
So we first need to calculate the mean.
That's a part of this need.
So to calculate that mean, just like we calculate any mean,
21 plus 23 plus 21 plus 22 plus 20 plus 19 plus 20 plus 22 plus
25 plus 24; we sum those,
we see that it equals 217 divided by 10 equals 21.7.
The mean for these data was 21.7.
This is the exact same mean we calculated
in the tutorial covering the mean.
So our mean value in our formula right here is 21.7.
So with that information,
we can continue computing the statistic.
So we take the first value, 21,
minus the mean 21.7 take the second value 23 minus 21.7,
third value 21 minus 21.7, 22 minus 21.7, 20 minus 21.7,
19 minus 21.7, 20 minus 21.7, 22 minus 21.7,
25 minus 21.7, 24 minus 21.7.
If we do this math, 21 minus 21.7 is negative .7,
23 minus 21.7 is 1.3, minus .7, .3 minus 1.7, minus 2.7,
minus 1.7, 0.3, 3.3, and 2.3.
And if we sum all of these differences,
you'll see that they sum to zero.
How do we get rid of that?
Well, we use the squared differences,
and that's why we square the difference negative .7 squared
equals .49, 1.3 squared equals 1.69, 49, .09, 2.89,
negative 2.7 squared is 7.29, again 2.89, .09,
3.3 squared is 10.89, and 2.3 squared is 5.29.
Now, if we sum those from the formula,
so we're summing the squared differences,
the sum of all of these is 32.1.
So we now have the numerator of our formula --
32.1 over n minus 1 is 10 minus 1 is 9; 32.1 divided 9 is 3.57.
The variance for these data is 3.57.
Again, this is the average squared difference.
The average squared difference; the variance 3.57.
So we've calculated our variance as 3.57.
Well, most of us do not think in terms of squared differences.
We think of average distance or average difference.
To get rid of the square,
we calculate the standard deviation,
which is just the square root of the variance.
So our variance is 3.57;
to calculate the standard deviation,
we just calculate the square root of 3.57 which is 1.89;
so our standard deviation is 1.89.
That represents the average distance from the mean.
The average distance from the mean.