This is Stephanie from StatisticsHowTo.com and in this video I will be showing you how
to find the population variance. Here is our formula for our population variance.
It is saying the sum of X minus mu, that is the mean, squared, all over N.
And I am going to break this down into steps for you,
so you can understand what all of these symbols mean and how to work the formula.
The first part of this equation I am going to work is the mean.
So, let say I had two sets of data. Say my first data set is 5 numbers; 28, 29,
30, 31, and 32. And my second data set also 5 numbers; 10,
20, 30, 40 and 50. So we want to get the mean for these numbers,
we are going to add the numbers together and divide by the number of items in the set.
28+29+30+31+32 is 150, and if I divide that by 5 I get a mean or average of 30.
For the second set of numbers, the mean is 10+20+30+40+50, that gives me
also 150 divided by 5, equals 30. First set of numbers they are very tightly
clustered around the mean. The mean is 30 right here, that we calculated
over here. 28, 29, 30, 31, and 32 are very close to 30.
But then this set of numbers where we calculated the mean to be 30.
10 and 50 are very far apart. So the mean tells you the average but it doesn’t
really tell you anything about the spread of data.
And that is where our population variance comes in.
The easiest way to work this formula is with a table.
So let me insert a table. I am going to find the population variance
for my first set of numbers that is 28, 29, 30, 31, and 32.
In my first column, I am just going to write my numbers.
First is my x-values; 28, 29, 30, 31, and 32.
And my second column, I am going to work this part of the formula right here, x minus mu,
the mean. So this is going to be x minus the mean, we
already figured out the mean right up here which is 30.
So 28 minus mu, minus 2. Going to work down the column, minus 1, 0,
1 and 2. We have worked out this part of this formula.
Then next I want to square these values. I am going to square the values in that column,
that would be x minus mu, and I am going to square that.
So what I am doing is I am squaring these values.
2 x 2 is a 4. I am working my way down, and this just gives
me a set of positive values. The next part of the formula is the sum of
x minus mu squared. What it is saying is add all of these values
in this column up. So my sum of x minus mu, squared is going
to be 4+1+1+4, that gives me 10. And finally the formula is telling us to divide
by N, the number of items in the set. So I am going to take this figure.
Sum of x minus mu squared, all divided by N.
So, 10 divided by N which is 5, that gives me 2.
That is my population variance for this first set of numbers.
I am just going to write up that here. This set one my population variance that is
sigma squared is equal to 2. Now I am going to work the same formula using
the same table for that second set of numbers. Again in my first column, put my x-values,
this time it is 10, 20, 30, 40 and 50. x minus mu, my x-value minus the mean, the
mean for the second data set was also 30. So 10 minus 30 is minus 20.
And we are working down the column, I’ve got minus 10, 0, 10 and 20.
The next column is asking us to square these values.
So 20 squared is 400. 10 squared is 100, and we have 0, a 100 again
and 400. We are almost there the next part is to add
all of these values up. So, 400+100+100+400 is 1000.
And the last part of the equation is asking us to to divide by N the number of items in
the data set. So we have 1000 divided by 5 items.
That gives us 200. So my population variance for my second data
set is 200. This 2 is a pretty small variance.
It tells you the values are tightly clustered around the mean, which is indeed true 28,
29, 30, 31, and 32. They are all very tightly clustered together.
And this large variance of 200 tells us that the numbers are pretty spread apart which
is also true. 10 and 50 are pretty far away from 30.
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