Now we've spoken about student's t-test and everything really follows from that. If you really understand how that works, how the central limit theorem works, how we are just converting everything to so many units of standard error, and what the area under the curve represents, you really understand most of statistical analysis. Now, student's t-test, and all the t-tests, really for comparing the means of two groups, but sometimes we want to compare the means of more than two groups. And that is when we have Anova, the most common one is called a one-way Anova. The one-way refers to the fact that we are only dealing with one factor. So imagine, we're just looking at comparing the means of white cell counts between four or five groups of participants. So it's just the white cell count, that's the only thing we're looking at, that is just one factor. And also has some levels to it that just really refers to how many groups we ask. If it's a four group, that's really a four level, one-way Anova if we look at that. Now t-test, remember we are comparing the means of two groups, but really over ANOVA three or more groups we are comparing to each other. Now all the assumptions we spoke about when we spoke about student's t-test has to be met. The ANOVA is still a parametric test, so we have to have those same assumptions. Most notably, the data points must come from, or any variable, must come from a variable that has a normal distribution in the patient or in the population. The actual data point values, they must be from numerical type and they must be a continuous data type. We also have to have roughly equal variances between the values in each group. If we work out the variances, standard deviation for each individual group, they've gotta be roughly the same. And also these participants really have to be independent. We can't be dealing with one is a called it triplets for instance, as an example. Now what the Anova does, it only determines that there's a difference between the groups. It doesn't highlight one group over any of the other groups, two groups over any of the other groups. It can not do that. It's only going to tell you that there's some fundamental difference between the groups but there's no individual group analysis that goes on. Now, it uses what we call the F-distribution. We haven't spoken about the F-distribution before, but it's a distribution, just as any other. Doesn't look normally distributed, though. It has this positive skew, so it's got this tail going out towards the right. And it is shaped by the degrees of freedom, just like the t-test, the T-distribution I should say, and it has a mean of around one. We're going to do exactly the same thing though, there's going to be some critical values. So many standard errors away from the mean would represent an area under the curve of open 0.05. And the differences between all of those groups is gonna be converted to an F-value and that F-value is gonna fall either side of this critical value. Everything would be the same. Here you see a beautiful example of random variables taken from, I think it was about 10,000 random variables, taken from an F-distribution. And you can beautifully see the mean is around one, but there is this tail that goes off to the right, there's quite a bit of skewness to the F-distribution. So what the Anova test really does, it compares the intergroup, the within group variability to the between group variability. And if it finds that the between group variability is much bigger than the intergroup variability, the variability of the data values within each group, then it really tends towards an f-value that'll be right far away from the critical value that will have statistical significance in some way. It's kind of the rule of thumb. Now, there are really many Anovas. You can well imagine from what I've discussed now, how many groups there are, how many values but different groups there are, how they interact with each other, how many factors we are dealing with is a lot of different types of the Anova. In essence though, we are comparing the means of more than two groups.