Shannon Entropy Calculator
Calculate information entropy to measure randomness and uncertainty in probability distributions
Calculate Shannon Entropy
Probabilities
Shannon Entropy Results
Very high entropy - maximum randomness achieved
Shannon Entropy Formula: H(X) = -∑ P(xi) × logb(P(xi))
Base: 2
Events: 2
Example Calculation
Sequence Analysis Example
Sequence: 1 0 3 5 8 3 0 7 0 1
Character frequencies:
- • '1': 2 occurrences (probability = 2/10 = 0.2)
- • '0': 3 occurrences (probability = 3/10 = 0.3)
- • '3': 2 occurrences (probability = 2/10 = 0.2)
- • '5': 1 occurrence (probability = 1/10 = 0.1)
- • '8': 1 occurrence (probability = 1/10 = 0.1)
- • '7': 1 occurrence (probability = 1/10 = 0.1)
Calculation (Base 2)
H = -∑ P(xi) × log₂(P(xi))
H = -[0.2×log₂(0.2) + 0.3×log₂(0.3) + 0.2×log₂(0.2) + 0.1×log₂(0.1) + 0.1×log₂(0.1) + 0.1×log₂(0.1)]
H = -[0.2×(-2.32) + 0.3×(-1.74) + 0.2×(-2.32) + 0.1×(-3.32) + 0.1×(-3.32) + 0.1×(-3.32)]
H ≈ 2.446 bits
Entropy Units
Bits (Shannons)
Base 2 logarithm
Most common in computing
Nats
Natural logarithm (base e)
Common in mathematics
Dits/Bans/Hartleys
Base 10 logarithm
Historical significance
Entropy Tips
Maximum entropy occurs when all events are equally likely
Zero entropy means perfect predictability (one outcome)
Higher entropy indicates more randomness and unpredictability
Probabilities must sum to 1.0 for valid entropy calculation
Use frequencies when you have count data instead of probabilities
Understanding Shannon Entropy
What is Shannon Entropy?
Shannon entropy, developed by Claude Shannon in 1948, is a measure of the average amount of information or uncertainty contained in a message or system. It quantifies how unpredictable or random the outcomes of a probability distribution are.
Applications
- •Information Theory: Data compression and communication
- •Ecology: Biodiversity and species distribution analysis
- •Machine Learning: Decision trees and feature selection
- •Cryptography: Password strength and randomness testing
Formula Explanation
H(X) = -∑ P(xi) × logb(P(xi))
Shannon Entropy Formula
- H(X): Shannon entropy of random variable X
- P(xi): Probability of outcome xi
- logb: Logarithm with base b
- ∑: Sum over all possible outcomes
Note: The negative sign ensures entropy is always non-negative, since log(P(xi)) is negative when 0 ≤ P(xi) ≤ 1.
Information Content
Measures the average information content per symbol in a message or data stream.
Uncertainty
Quantifies the unpredictability of outcomes in a probability distribution.
Diversity Index
Used in ecology and other fields to measure diversity and evenness of distributions.