ANOVA 1: Calculating SST (total sum of squares) Analysis of Variance 1 - Calculating SST (Total Sum of Squares)
ANOVA 1: Calculating SST (total sum of squares)
- In this video and in the next few videos we'll actually be doing a bunch of calculations about this data set right over here.
- And hopefully just going through those calculations will give you an intuitive sense of what the analysis
- of variance is all about. Now the first thing I wanna do in this video is calculate the total sum of
- squares. So I call that 'SST'. Sum of squares total. And you could view it as really the numerator when
- you calculate the variance. So we're just gonna take the distance between of each of these data points
- and the mean of all of these data points, square them and just take that sum, we'll not really divide
- by the degree of freedom, which you normally do if you're calculating sample variance.
- Now what is this going to be? Well the first thing we got to do is we have to figure out the mean of
- all of this stuff over here. And I'm actually gonna call that the grand mean.
- I'm gonna call that the grand mean. And let me show you in a second that it's the same thing as the mean
- of the means of each of these data sets. So let's calculate the grand means.
- So that's gonna be 3 plus 2 plus 1. 3 plus 2 plus 1 plus 5 plus 3 plus 4 plus 5 plus 6 plus 7 ... plus 5 plus 6 plus 7.
- And then we have nine data points here. We have nine data points so we're gonna divide by nine and then
- this is gonna be equal to '...'. 3 plus 2 plus 1 is 6. 6 plus, let me just.. er so these are 6. 5 plus
- 3 plus 4 is, that's 12. And then 5 plus 6 plus seven is 18. And then 6 plus 12 is 18, plus another 18 is 36
- divided by nine is equal to 4. Let me show you that that's the exact same thing as the mean of the means.
- So this, the mean of this group one over here, that's seen in green, the mean of group one over here
- is 3 plus 2 plus 1, that's 6 right over here, divided by 3 data points, so that would be equal to 2.
- The mean of group 2, the sum here is 12, we saw that right over here: 5 plus 3 plus 4 is 12, divided
- by 3 is 4, cause we have three data points. And then the mean of group 3, 5 plus 6 plus 7
- is 18 divided by 3 is 6. So if you're gonna take the mean of the means
- which is in another way this grand mean, you have 2 plus 4 plus 6
- which is 12 divided by 3 means here and once again you'd get 4.
- So you can view this the mean of all of the data and all of the groups
- or the mean of the means of each of these groups. But either way now that we've calculated it
- we can actually figure out the total sum of squares. So let's do that.
- So it's going to be equal to: 3 minus 4, the 4 is this 4 right over here, squared plus
- 2 minus 4 squared plus 1 minus 4 squared, now I'll do these guys over here in purple,
- plus 5 minus 4 squared plus 3 minus 4 squared plus 4 minus 4 squared
- I'll just scroll over here a little bit, plus 4 minus 4 squared. Now we only have three left.
- Plus 5 minus 4 squared plus 6 minus 4 squared plus 7 minus 4 squared. Now what does this give us?
- So up here this first is gonna be equal to, 3 minus 4 the difference is 1, you square it,
- you're gonna get, er, it's actually a negative 1, you square it you get one.
- Plus, you get negative 2 squared is 4 plus negative 3 squared. Negative 3 squared is 9.
- And then we have here in the magenta: 5 minus 4 is 1, squared is still 1. 3 minus 4 squared is 1 you
- square it again you still get 1 and 4 minus 4 is just a 0. So we can ... let me just write a 0 here
- just to show you that we actually calculated that. And then we have these last 3 data points.
- 5 minus 4 squared, that's one. 6 minus 4 squared, that is 4, it's 2 squared. And then plus 7 minus 4 is 3
- squared is 9. So what's this going to be equal to. So I have 1 plus 4 plus 9.
- 1 plus 4 plus 9 right over here, that's 5 plus 9. This right over here is 14, right?
- 5 plus ..., yep, 14. And we also have another 14 right over here cause we have a 1 plus 4 plus 9
- so that right over there is also 14. And then we have 2 over here. So it's gonna be
- 28, 14 times 2, 14 plus 14 is 28, plus 2 is 30. Is equal to 30. So our total sum of squares
- And actually if we wanted the variance here we would divide this by the degrees of freedom.
- And these are multiple times the degrees of freedom here. So let's say, let's say that we have
- so we know we have m groups over here, so let me just write this m. And, I'm not gonna
- prove things rigorously here but I want you to show, I wanna show you
- where some of these strange formulas that show up in statistics would actually come from
- without proving it rigorously, more to give you the intuition. So we have m groups here
- and each group here has n members. So how many total members do we have here?
- Well we have m times n or 9, right? 3 times 3 total members. So degrees of freedom, we remember, you
- have this many, however many data points you have minus 1 degrees of freedom. Because if you know
- if you knew the mean of means, if you know the mean of means, if you assume you knew that
- then you only would, would only n, only, er, hehe. 9 minus 1, only 8 of these were going to give you
- new information because if you know that you could calculate the last, or it really wouldn't have to be the
- last one if you have the other 8 you can calculate this one. If you have 8 of them you could always calculate
- the 9th one using the mean of means. So one way to think about it is that theres only 8 independent
- measurements here. Or if you want to talk in terms of general, you want to talk in general, there are
- m times n, so that is total number of samples, minus 1 degrees of freedom.
- And if you're actually calculating the variance here we would just divide 30 by m times n minus 1.
- Or this is another way of saying 8 degrees of freedom for this exact example. You take 30
- divided by 8 and you actually have the variance for this entire group, the group of 9 [...].
- I'll leave you here in this video. In the next video we're gonna try to figure out how much of this total
- variance, how much of this total squared sum, total variation, comes from the variation
- within each of these groups versus the variation between the groups. And I think you'll
- get a sense of where this whole analysis of variance is coming from. Look there is the variance
- of this entire sample of nine but some of that variance, if these groups are different in some way,
- might come from the variation from being in different groups versus the variation from being within a
- group. We're gonna calculate those two things and we're going to see that they're going to add up
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