When we collect raw data and we plot its frequency, we're never going to see a nice normal distribution, a bell-shaped curve. There's going to be some abnormality and asymmetry. That is very common. Now, we have two concepts, two terms, that describes this asymmetry in the values that we do collect, and one is skewness and the other is kurtosis. Now skewness is the easiest to understand, and the term really says it all. There is some skewness in the data, there isn't symmetry. Look at this first graph. Look at the histograms, those little rectangles. The vast majority of the values occurred on the upper end of the values for that variable. And it sort of tails off towards the negative side, that's towards the left-hand side. That is negative skewness. And the next one you'll see is positive skewness. We have a tail that goes out to the right. The vast majority of values occur on the smallest side, so that's skewness very easy to understand. Kurtosis is a bit difficult. There's a bit more difficult to understand and there really isn't a universally accepted definition, although I think most statisticians know we on a proper definition, but really, there is still some ambiguity in its interpretation. And the newest concept really is about what happens in the shoulders. Now, the shoulder of a distribution curve is that area between the little spike at the top, and the little tails. That'll be the shoulder. And I'll show you something about that. What I want us to concentrate on on the course though, is the thickness of the tails on both sides, the thickness of those two tails. Now, we do define two types of kurtosis. One is a Platykurtic curve and one is Leptokurtic curve. Be very careful of those terms that they really are define for a a draft that has a skewness of zero. And we know we hardly ever going to see at least in our raw data that kind of skewness of zero. But let's assume that. Then we do get these two curves, the platykurtic curve and the leptokurtic curve. And the platykurtic curve is really flatter. It has a broader peak and pushes more of the values out into the tails, gives us those fatter tails. And for the same T statistic, we're either going to get a high A under the curve or a higher P value. And then, the leptokurtic curve which is more peaked, move those values up near the middle and there are less values out on the tail. But let me show you this concept of the shoulder, look at these two graphs, one on the top, one on the bottom. The top one we see the graph in red which is more leptokurtic and the graph in the black which is more platykurtic, in other words, the red one is more spiked, more of the values are centered around the mean of 0 there. And you can see if we go out on the red graph towards the tail, the leptokurtic a few values out from the tail, this is the more platykurtic curve. Its peak is a bit lower, and there are more values pushed out into the tails. But look at the graph for the orange and the blue. And graphs, the blue one there you see spikes a lot more that is a leptokurtic curve but look at the tails. Certainly for some section there are more values in the tails than the more platykurtic or orange curve. It has a flatter peak but it has very few in the tail. Look at the difference in the shoulders there, with the orange curve you'll find more values built into the shoulders. So you can see two beautiful examples there of why it is so difficult to really interpret kurtosis. I don't want us to spend too much time on kurtosis though. We are going to deal with a normal distribution, and we're going to deal with a T distribution, and then eventually non-parametric distributions. We're not really concerned about the kurtosis in our distributions. Purely because of the fact that in health based statistics we really want enough patients, or enough subjects, in a trial or in a study, that we really end up with a nice, rather nice T distribution or some form of T distribution. And we're not that concerned about. But understand these concepts of skewness and kurtosis, and be slightly circumspect when you interpret that kurtosis rather have those graphs drawn out and interpret values from those graphs. And understand when you do see these when you have fatter tails and when you have these thinner tails, depending on the shape of those shoulders.