Shannon Entropy Calculator – Information Entropy
Calculate information entropy and uncertainty in probability distributions
How to Use
- Enter probability values for each event (must sum to 1)
- Add or remove probability fields as needed
- Click calculate to compute Shannon entropy
- View entropy, maximum entropy, and normalized entropy
What is Shannon Entropy?
Shannon Entropy, introduced by Claude Shannon in 1948, is a measure of the average information content or uncertainty in a probability distribution. It quantifies how much information is needed, on average, to describe the outcome of a random variable.
The formula for Shannon Entropy is H(X) = -Σ(p(x) × log₂(p(x))), where p(x) is the probability of event x. Entropy is measured in bits when using log base 2.
Interpreting Entropy Values
Shannon Entropy ranges from 0 to log₂(n), where n is the number of possible events:
- Minimum entropy (0 bits): Occurs when one event has probability 1 and all others have probability 0 - complete certainty, no uncertainty
- Maximum entropy (log₂(n) bits): Occurs when all events are equally likely - maximum uncertainty and randomness
- Intermediate values: Indicate partial uncertainty in the distribution
Normalized entropy divides entropy by maximum entropy to give a value between 0 and 1, making it easier to compare distributions with different numbers of events.
Applications of Shannon Entropy
- Information Theory: Measuring information content in data transmission
- Data Compression: Determining optimal compression algorithms
- Machine Learning: Feature selection and decision tree construction
- Cryptography: Assessing randomness and security of encryption
- Biology: Analyzing genetic diversity and protein sequences
- Economics: Measuring market diversity and portfolio risk
- Natural Language Processing: Analyzing language patterns and predictability
Shannon Entropy Examples
Consider these example probability distributions:
| Distribution | Probabilities | Entropy | Interpretation |
|---|---|---|---|
| Coin flip | [0.5, 0.5] | 1 bit | Maximum entropy for 2 events |
| Biased coin | [0.9, 0.1] | 0.47 bits | Lower entropy - more predictable |
| Fair die | [1/6, 1/6, 1/6, 1/6, 1/6, 1/6] | 2.58 bits | Maximum entropy for 6 events |
| Certain outcome | [1.0, 0.0] | 0 bits | Minimum entropy - no uncertainty |
Frequently Asked Questions
- What does Shannon Entropy measure?
- Shannon Entropy measures the average information content or uncertainty in a probability distribution. Higher entropy indicates more uncertainty and randomness, while lower entropy indicates more predictability and order.
- Why is it measured in bits?
- When using logarithm base 2 (log₂), entropy is measured in bits because it represents the average number of binary questions (yes/no) needed to determine the outcome. Using natural logarithm (ln) gives entropy in nats.
- What is maximum entropy?
- Maximum entropy occurs when all events are equally likely. For n events, maximum entropy is log₂(n) bits. This represents maximum uncertainty where no outcome can be predicted better than any other.
- How is Shannon Entropy used in machine learning?
- In machine learning, Shannon Entropy is used for feature selection, measuring information gain in decision trees, and evaluating model uncertainty. It helps identify which features provide the most information for classification tasks.
- Can entropy be negative?
- No, Shannon Entropy is always non-negative. The minimum value is 0 (complete certainty), and it increases with uncertainty. The maximum depends on the number of possible events.