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Shapes of Distribution

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This tutorial talks about shapes of distribution. Now, a distribution is a description of how all possible values of a variable occur. And you can show off the distribution in many ways. You can show it off with a frequency table, like we have here. You can use a graph like a histogram, or you can use a smoothed-out graph like this. You could also use some sort of mathematical rule to describe your distribution as well.

Now, one kind of distribution is a symmetric distribution. In a symmetric distribution, the mean and the median are the same, and you can draw a line of symmetry through the middle and create mirror halves. So in this first example here, with this distribution, the line of symmetry would be right around here. So we're cutting the data in half, and then we have mirror images.

Now, the mean and the median are going to be the same, and they're going to be right along this line symmetry, right here in this example. Over here, even though we have two modes, we can still have a symmetric distribution. Our line of symmetry is going to be right there-- creating those mirror images again.

Now, a special kind of symmetric distribution is a uniform distribution. The graph looks very flat. So if you look across the top here, it's all flat. And that's because all of the values have equal frequency. So this min and this min, and this and this and this and this are all the same height.

Now, in a uniform distribution, again, because it's symmetric distribution, the mean and the median are going to be the same. And if you draw a line of symmetry down the middle, you end up having mirror image halves. Now, if you have a non-symmetric distribution, we have a skewed distribution.

Because a skewed distribution is not symmetric, the mean is not the same as the median. With a skewed distribution, you have a tail of values that are going to be above or below the median. So this tail is pulling the mean towards it, and pulling the mean away from the median. We'll see what I mean in this next slide.

So for a skewed distribution, we can have two kinds. We can talk about left skew, or you can talk about right skew. And then other words that we can use to describe left skew is negatively skewed, and for right skew we can talk about being positively skewed. So here's what we're talking about.

In a left-skewed distribution, the mode is going to be that highest point. And this tail-- this shaded-in reading here-- is pulling the mean away from the median and away from the mode. So this tail, here, is below the median and below the mode. And the tail is on the left side, for a left skew.

Now, on the other hand, we have a right-skewed distribution, where the tail is going to be on the right side. So the same issue here-- this tail on the right side is pulling the mean away from the median so that the mean and the tail both end up being above the median. Let's do some practice identifying left versus right-skewed distributions.

So here, if we have an image like this, if we imagine smoothed-out version of this, our tail is going to be over here on the right, so this is a right-skewed distribution. In an example like this, our tail is again on the right, so this is going to be another example of a right-skewed distribution. Now, with an example like this, we have our tail coming to the left, so this is a left-skewed distribution. So in this example, the mean is below the median. In this example, the mean is above the median, and the mean is above the median. And I don't know exactly where the mean and median are in these examples, but we do know that the tail is pulling the mean away from the median. This has been our tutorial on the shapes of distributions.

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This tutorial talks about unimodal and bimodal and multimodal distributions. Unimodal distribution is also called "single-peaked." And the name comes from the look of the graph. If you look here, we can see that this graph has a single peak, a single highest point. So a unimodal distribution is a distribution that has one mode, and that is this bar here. This is the mode for this distribution.

Now the name kind of is giving us that clue. "Uni" means "one," like a unicycle has one wheel, and model is talking about the mode. So unimodal, one mode. Bimodal, you can probably guess that prefix "bi" is telling us two-- bicycle, two wheels, bimodal, two modes, two peaks.

And here's an example of that. We have two kind of high areas, two modes, two peaks. And even this peak doesn't get quite as high as the previous one, we can still consider that a local high. So we consider that a second peak.

Now multimodal, the "multi," multiple, "many," so we have more than two peaks. So anything like this, where as long as we have more than two peaks, more than two modes, we're talking about a multimodal distribution. We have a set of examples, and we're going to sort them in.

So with this example here, we can see two clear peaks. So I would call this one a bimodal distribution. Now this little bump down here could be considered a local high, so you might want to call that multimodal. But it really looks like it's probably bimodal. So when we're choosing between unimodal, bimodal, and multimodal, there's a little bit of leeway. This example here, again, has one peak, and then another peak here. So this would also be a bimodal distribution.

With this histogram, there is only one peak. There's one highest bar, so there is one mode. So this is unimodal. And in this example here, same thing, there is one peak. So this is unimodal. Now an example of multimodal would look like this, where we have a lot of high points, a lot of modes. We have definitely more than two, so this would be multimodal.

So this has been your tutorial on unimodal versus bimodal distribution. Unimodal has one peak. Bimodal has two peaks. Multimodal has more than two peaks.