Скачать или смотреть video 10.5. ANOVA sums of squares OUTDATED

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I already explained how to do an ANOVA, where you calculate the variance estimate for the top and the bottom of the ratio. It is pretty simple from your sample data, but there is a very different approach called the sums of squares (or SS) approach to doing ANOVAs. The way I showed you before only works with sample sizes that are the same across all three of your groups. But the sums of squares approach works with any kind of unequal group sizes, although it is really not intuitive how your math relates to the F ratio. But I wanted to do a really simple example to illustrate how it works.

What I have done here is I have the source table ready to fill in, and with the sums of squares approach this is kind of you start, with a source table and you fill it in as you go. Then, what I have done is I have created four different distributions. These are not set up the way they usually are. This is actually going to be our our observed sample data, and for this I am only going to have two groups. For each group, I am actually only gonna have two people. So, there is gonna be N equals 2 for both. The reason I am doing that, is the sums of squares approach is computationally intensive, so I am just going to have a small number of data points so that we don't have to spend a lot of time just going through the calculations, and when you have fewer data points, it is actually easier to visually understand what is happening a little bit better.

For this we have got our Group 1, it has two people in it. One person has got a 1. Another person has a 3. For our Group 2, one person has a 5. The other person has a 7. So, our total sample size is 4. Our Group 1 has a one and a [three]. And our Group 2 has a [five] and a seven. With the sums of squares approach, we are actually going to directly calculate the sums-of-squares between the sums-of- squares within, and the sums-of-squares total. That is what these three distributions are for here.

For this one, we are going to start with the sums of squares total. First, I am just gonna redraw our 4 data points and end up doing that for all of these. So, I have just drawn all the exact same data points that were up on this distribution onto each of these.

We are going to start down here. We are gonna work on the sums of squares total. The sums of squares total ignore grouping all together. Remember, this is a sums of squares, so it is the sum of the squared differences. Remember, this is the top of our variance formula. So, we are gonna calculate the sums of squares ... we are gonna calculate the sums of squares overall. That will go in right here. The first thing we need for calculating the sums of squares total is the mean for all of these points. We will call that the "grand mean," because it ignores grouping. We have a 1, plus a 3, plus a 5, plus a 7. And, there are 1, 2, 3, 4 points. So, this is 10 ... 16, divided by 4, equals 4. So, our grand mean is gonna equal 4. I can put that 4, that grand mean, right into our formula here, because that will be the same for our sums of squares total. Sums of squares total equals, our first score, 1, minus the grand mean, squared, plus 3 minus the grand mean, squared, plus 5 minus the grand mean, squared, plus 7 minus the grand mean, squared. -3 squared, plus -1 squared, plus 1 squared, plus 3 squared. 9 + 1 +1 + 9. So, our sums of squares total, it is gonna be 20. So, the sums of squares total, we could actually can visualize what is happening here. So, our grand mean, it was 4. Right up here. Then, what we did is we took this score here, we calculated 1 minus 4. We got -3, and then we squared it, and that was 9. Then, we took this score here, and we subtracted the mean from it. We got -1, squared. That gave us 1. For this one, we took 4 from 5, and we got 1. We squared that, and we got 1. For the 7 over here, we got 7 - 4, was 3, squared, equals 9. So, we just add them up to sum them. So we sum up 9, 1, 9, and 1. We get 20. That is our sums of squares for our sum squares total.

For our degrees of freedom total, remember that is just the total sample size, ignoring grouping, minus one. In this case, we have one, two, three, four people, minus 1. Our degrees of freedom total, it is gonna be 3.
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