Brillouin Index (HB)

Understanding Brillouin's Diversity Index (H_B)

Brillouin's Index (H_B) is a measure of species diversity that is particularly suited for collections where all individuals have been identified and counted (i.e., a fully censused community, not a random sample from a larger population). It is based on information theory and considers the total number of ways individuals can be arranged among species.

Formula:

HB = (1 / N) * [ ln(N!) - Σ ln(ni!) ]

• N is the Total Number of Individuals of all species in the collection.
• ni is the number of individuals in species i.
• S is the Species Richness (total number of different species present).
• Σ represents the sum across all S species.
• ln is the natural logarithm (log base e).
• k! denotes the factorial of k (e.g., 5! = 5 × 4 × 3 × 2 × 1).
• The term ln(k!) can be calculated as Σj=1 to k ln(j). By convention, ln(0!) = ln(1) = 0.

Interpretation:

• Higher H_B values indicate greater diversity.
• If all individuals belong to one species (S=1, n₁=N), then H_B = 0.
• The index is sensitive to both species richness (S) and the evenness of individual distribution among species.

Important Notes:

• This index is used when the collection is finite and fully known, not for samples where estimation of a larger community's diversity is the goal (Shannon's index is often preferred for samples).
• Calculating factorials directly can lead to very large numbers. This calculator uses the property ln(k!) = Σ ln(j) to avoid overflow issues.
• The index is undefined or 0 if N=0. If N=1, HB will be 0.

How to Use:

1. Enter the number of individuals (count) for each species identified in your collection.
2. Use the "Add Species" button to create more input rows as needed.
3. Click "Calculate Brillouin Index" to see the results.