SME
Grade 9 Science · Problem type
Simpson’s diversity index
Use this for “Which pond community is more diverse: 25, 5, 5, 5 or 10, 10, 10, 10?”
Chart
25 + 5 + 5 + 5 = 40.
This is an example. Type your own values to start a new problem.
S
=
1
−
n₁
N
2
−
n₂
N
2
−
n₃
N
2
−
n₄
N
2
Formulas
N
=
n₁
+
n₂
+
n₃
+
n₄
40 = 25 + 5 + 5 + 5
S
=
1
−
(
(
n₁
N
)
2
+
(
n₂
N
)
2
+
(
n₃
N
)
2
+
(
n₄
N
)
2
)
0.563
=
1
−
(
(
25
40
)
2
+
(
5
40
)
2
+
(
5
40
)
2
+
(
5
40
)
2
)
Assumptions
•
S is the chance that two picks at random, the first put back, belong to different species.
•
More species, and more even counts of each, raise S toward 1; one species alone gives 0.
•
Here a pond survey counts four species.
Step-by-step
WE KNOW
n₁ = 25, n₂ = 5, n₃ = 5, n₄ = 5
FIND
individuals in all (N), diversity index (Simpson’s) (S)
Step 1 · Find individuals in all (N)
Use N = n₁ + n₂ + n₃ + n₄
Add the counts of the four species.
N
=
25
+
5
+
5
+
5
N
=
40
Step 2 · Find diversity index (Simpson’s): S
Use S = 1 − ((n₁ ÷ N)² + (n₂ ÷ N)² + (n₃ ÷ N)² + (n₄ ÷ N)²)
Each
(
n
N
)
2
is
the
chance
two
picks
are
both
that
species;
1
minus
their
sum
is
the
chance
they
differ.
S
=
1
−
(
(
25
40
)
2
+
(
5
40
)
2
+
(
5
40
)
2
+
(
5
40
)
2
)
S
=
1
−
(
0.625
2
+
0.125
2
+
0.125
2
+
0.125
2
)
S
=
1
−
(0.3906
+
0.01563
+
0.01563
+
0.01563)
S
=
1
−
0.4375
S
=
0.563
Check
Put the numbers back in. Both sides should match.
40 = 25 + 5 + 5 + 5 ✓
0.563
≈
1
−
(
(
25
40
)
2
+
(
5
40
)
2
+
(
5
40
)
2
+
(
5
40
)
2
)
✓
Topic check
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