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: Write the working formula.
Simpson's index is
\[ D = 1 - \sum_i p_i^2, \quad p_i = \frac{n_i}{100} \] so
\[ D = 1 - \frac{\sum_i n_i^2}{10000} \]

Step 2: Square and sum the counts for Community P.
The counts for P are \(69, 18, 5, 4, 1, 1, 0, 1, 0, 1\). Squaring and adding:
\[ 69^2+18^2+5^2+4^2+1^2+1^2+0^2+1^2+0^2+1^2 = 4761+324+25+16+1+1+0+1+0+1 = 5130 \]
So
\[ D_P = 1-\frac{5130}{10000} = 1-0.513 = 0.487 \]

Step 3: Repeat for Community Q.
The counts for Q are \(45, 43, 4, 3, 1, 1, 0, 1, 0, 1\). Squaring and adding:
\[ 45^2+43^2+4^2+3^2+1^2+1^2+0^2+1^2+0^2+1^2 = 2025+1849+16+9+1+1+0+1+0+1 = 3903 \]
So
\[ D_Q = 1-\frac{3903}{10000} = 1-0.3903 = 0.6097 \]

Step 4: Repeat for Community R.
The counts for R are \(70, 21, 4, 5, 0, 0, 1, 0, 1, 0\). Squaring and adding:
\[ 70^2+21^2+4^2+5^2+0^2+0^2+1^2+0^2+1^2+0^2 = 4900+441+16+25+0+0+1+0+1+0 = 5384 \]
So
\[ D_R = 1-\frac{5384}{10000} = 1-0.5384 = 0.4616 \]

Step 5: Compare the three values pairwise.
The three index values are \(D_P = 0.487\), \(D_Q = 0.6097\), and \(D_R = 0.4616\). The differences are
\[ |D_P-D_Q| = 0.123, \quad |D_P-D_R| = 0.025, \quad |D_Q-D_R| = 0.148 \]

Step 6: Pick the closest pair.
The smallest difference is between \(D_P\) and \(D_R\), so P and R have the most similar diversity index. Q stands apart with a clearly higher index than both.

Final Answer:
P is more similar to R than either of them is to Q.
\[ \boxed{\text{Option (B)}} \]