SME
Human Geography · Problem type
Language or religious diversity
Use this for “A country speaks four languages: 50 %, 30 %, 15 % and 5 % of people. What is its diversity index?”
Chart
50% + 30% + 15% + 5% = 100% of the whole.
This is an example. Type your own values to start a new problem.
Σ
=
(
s₁
2
+
s₂
2
+
s₃
2
+
s₄
2
)
÷
10,000
F
=
1
−
Σ
Formulas
s₄
=
100
−
s₁
−
s₂
−
s₃
5 = 100 − 50 − 30 − 15
Σ
=
s₁
2
+
s₂
2
+
s₃
2
+
s₄
2
10,000
0.365
=
50
2
+
30
2
+
15
2
+
5
2
10,000
F
=
1
−
Σ
0.635 = 1 − 0.365
Assumptions
•
F is the chance that two people picked at random speak different languages or follow different religions.
•
Σ is the chance both belong to the same group: each share squared, summed, as a fraction.
•
With four groups, F runs from 0 (everyone in one group) to 0.75 (four equal groups).
Step-by-step
WE KNOW
s₁ = 50%, s₂ = 30%, s₃ = 15%
FIND
group 4 share (s₄), chance of the same group (Σ), diversity index (F)
Step 1 · Find group 4 share (s₄)
Use s₄ = 100 − s₁ − s₂ − s₃
The four shares make up everyone, 100 %, so the last group is what the other three leave.
s₄
=
100
−
50
−
30
−
15
s₄
=
5%
Step 2 · Find chance of the same group (Σ)
Use Σ = (s₁² + s₂² + s₃² + s₄²) ÷ 10,000
Square each share and add; dividing by 10,000 (100 × 100) turns the percents into a fraction.
Σ
=
50
2
+
30
2
+
15
2
+
5
2
10,000
2,500 + 900 + 225 + 25 = 3,650
3,650
10,000
=
0.365
Σ
=
0.365
Step 3 · Find diversity index (F)
Use F = 1 − Σ
Two
people
are
either
in
the
same
group
or
not,
so
the
chance
they
differ
is
1
minus
Σ
.
F
=
1
−
0.365
F
=
0.635
Check
Put the numbers back in. Both sides should match.
5 = 100 − 50 − 30 − 15 ✓
0.365
=
50
2
+
30
2
+
15
2
+
5
2
10,000
✓
0.635 = 1 − 0.365 ✓
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