Question

The table below gives the abundance of species 1 to 10 in three communities, P, Q, and R, each with a total of 100 individuals. Simpson's diversity index for a community is
\[ D = 1 - \sum_i p_i^2, \] where \(p_i\) is the proportion of individuals in the community belonging to the \(i^{th}\) species.

Species	Community P	Community Q	Community R
1	69	45	70
2	18	43	21
3	5	4	4
4	4	3	5
5	1	1	0
6	1	1	0
7	0	0	1
8	1	1	0
9	0	0	1
10	1	1	0

Which one of the following statements is true about the comparison of Simpson's index between these communities?

• A. P is more similar to Q than either of them is to R
• B. P is more similar to R than either of them is to Q
• C. Q is more similar to R than either of them is to P
• D. P, Q, and R are equally similar to each other

Hint:

Work out \(D=1-\Sigma p_i^2\) for each community and compare, or notice that P and R are both dominated by one species while Q is more even.

Answer 1

The Correct Option is B

Step 1: Look at how each community is dominated before doing any sums.
In community P, one species (species 1) makes up 69 out of 100 individuals, and a second species (species 2) adds another 18. So the top two species already cover 87 individuals, leaving very few for the rest.
In community R, the pattern is almost the same: species 1 has 70 and species 2 has 21, again covering 91 of the 100 individuals.
In community Q, however, species 1 has only 45 and species 2 has 43, so the two top species are close in size and together cover 88, but no single species dominates the way it does in P or R.

Step 2: Connect dominance to the diversity index.
Simpson's index falls when one species dominates, because a big \(p_i\) contributes a large \(p_i^2\) term. Since P and R are both heavily dominated by a single species in a similar way, their index values should sit close together and lower than Q's. Q, being more even between its top two species, should carry a noticeably higher index.

Step 3: Confirm with the actual numbers.
Working out $D = 1 - \dfrac{\sum n_i^2}{10000}$ for each community gives
\[ D_P = 0.487, \quad D_Q = 0.610, \quad D_R = 0.462 \]
This matches the pattern spotted from the raw counts: Q sits well above the other two, while P and R are close together, only about $0.025$ apart.

Step 4: Conclude.
Since P and R have the two closest Simpson's index values, they are the most similar pair.
\[ \boxed{\text{P is more similar to R than either of them is to Q}} \]