Simpson's index of diversity

- [Instructor] So in this table here we have two different communities, community one and community two, and each of them contain three different species. And we see the populations of those three different species. And we also see that the total number of individuals in each community is the same. They both have a total of 1,000 individuals. Now, my question to you, just intuitively based on the data in this table, which community would you say is more diverse and why, community one or community two? All right, now let's think about this together. So as we already talked about, they have the same number of individuals, and you might be thinking that the number of species could be related to the diversity, and you'd be right. The number of species does contribute to the diversity, but we're dealing with a situation where both communities have the same number of species. They each have three species. But when we look at the data, it's clear that community two is mostly species A and you have very small groups of species B and species C, while community one is more evenly spread. So just intuitively it feels like community one is maybe more diverse, but this was just on my intuition or our intuition, and the numbers are pretty clear here. It's evenly distributed amongst the species here, and here it's very heavily weighted on species A, but it might not always be this clear. So it'd be useful to have some type of quantitative way to measure the diversity of a population. And lucky for us, there is a quantitative way to do that called Simpson's, I'll write it down, Simpson's diversity index, and the way you calculate it, it's equal to one minus the sum of, for each species you take the number of that species divided by the community size squared. So for each of the species, you do this calculation, square it and then you add it up for each of those species. So let's figure out Simpson's diversity index for both communities one and community two. And I encourage you you could pause the video and try to work on it on your own before I work through it with you. So let's start with community one. So I'll say diversity index for community one. I'll just put that in parentheses, is going to be equal to one minus, so we have 325 over, over 1,000 squared. Remember, we're gonna sum on each of these species plus 305, 305 over 1,000 squared plus 370 over 1,000 squared. And I need to close my parentheses, and I can simplify this a bit. This is going to be equal to one minus, so all of these thousand squares, a thousand squared is a million. So it's gonna be everything over 1 million, 1 million, and then we're going to have 325 squared, plus 305 squared, plus 370 squared. And that is going to give us 325 squared, plus 305 squared, plus 370 squared is equal to that. That's the numerator here, and I'm willing to divide that by a million divided by, one, one, two, three, one, two, three. And that is a million. It equals this. And then I'm gonna subtract that from one. So let's put a negative sign here and say, plus one is equal to 0.664. So this is going to be approximately equal to 0.6664. Now let's do the same thing for community two. So if I write it over here, the diversity index for community two is going to be equal to one minus, I put a big parenthesis here, and we're going to have 925 over 1,000 squared, plus 40 over 1,000 squared, plus 35 over 1,000 squared. And if we simplify in a similar way, that's gonna be equal to one minus all these thousand squares. That's just a million, and that's a common denominator. And so you're gonna have 925 squared, plus 40 squared, plus 35 squared. And then this is going to be approximately equal to 925 squared, plus 40 squared, plus 35 squared is equal to this divided by a million. So divided by one, one, two, three, one, two, three. Yep, six zeros is equal to that. And then you subtract that from one and you get, which is approximately equal to 0.142. And so we see very clearly when we use Simpson's diversity index that consistent with our intuition community two has a lower diversity index than community one. And it's consistent with our intuition that it is less diverse. And I encourage you after this video think about why that makes mathematical sense.