Shannon Entropy Calculator

What This Shannon Entropy Calculator Does

This calculator computes the Shannon entropy of a discrete probability distribution. You enter a list of probabilities, and it returns how much uncertainty or average information is contained in one draw from that distribution, measured in bits.

Shannon entropy is central to information theory, data compression, coding, cryptography, and many machine learning methods. A higher entropy value means a more unpredictable source; a lower entropy value means more regularity and easier compression.

Shannon Entropy: Definition and Formula

Consider a discrete random variable with possible outcomes . Each outcome occurs with probability . The Shannon entropy of this variable is defined as:

$$H=-\sum _i{p}_i{log}_2{p}_i$$

In standard mathematical notation, this is often written as:

H = - Σ p_i log2(p_i)

The logarithm is taken in base 2, so the result is measured in bits. One bit corresponds to the information gained from observing the outcome of a fair binary choice (such as an idealized coin flip).

How to Use This Calculator

1. Prepare your probabilities: Identify all distinct outcomes of your discrete random variable and their probabilities. Each probability should be a number between 0 and 1.
2. Format the input: Enter the probabilities as comma-separated values, for example:
   0.5, 0.5 or 0.1, 0.2, 0.3, 0.4.
3. Submit the form: Click the button to compute the Shannon entropy in bits.
4. Read the result: The calculator reports a single numeric value. Higher values correspond to greater uncertainty and higher average information per outcome.

Usage Assumptions and Input Requirements

When you use this Shannon entropy calculator, keep the following assumptions and practical details in mind:

• Discrete distributions only: The formula and this tool apply to discrete outcomes (e.g., categories, symbols, classes). It is not intended for continuous probability density functions.
• Non-negative probabilities: Each value you enter should be between 0 and 1 inclusive. Negative numbers are not valid probabilities.
• Handling of zeros: Terms with probability 0 are treated as contributing 0 to the entropy, because the limit of as is 0.
• Sum-to-one expectation: Conceptually, the list of probabilities should sum to 1. If your probabilities represent relative frequencies or unnormalized weights, you should convert or normalize them to probabilities before interpreting the entropy as "bits per outcome" in the strict Shannon sense.
• Numerical rounding: Internally, entropy is computed with finite precision arithmetic. Results may be rounded for display.
• Interpretation context: The entropy value describes uncertainty in one draw from the specified distribution, assuming independent and identically distributed repeated draws.

If the values you enter do not satisfy these assumptions, you can still obtain a numeric output, but its interpretation as true Shannon entropy may be misleading.

Interpreting the Entropy Result

The numeric value returned by the calculator has a clear meaning:

• Units: The entropy is in bits, because the logarithm is base 2.
• Zero entropy: An entropy of 0 bits means complete certainty. One outcome has probability 1 and all others have probability 0. Observing the outcome conveys no new information, because it is fully predictable.
• Higher entropy: Larger values indicate more unpredictability. Outcomes are more evenly spread and each observation reveals more information about which outcome occurred.
• Maximum entropy for fixed : For possible outcomes that are all equally likely (each with probability ), the entropy is maximal and equal to bits. This situation represents the highest possible uncertainty for a system with discrete outcomes.

For example, a fair coin has 2 outcomes with equal probabilities, so its maximum entropy is bit. A fair six-sided die has bits of entropy per roll.

Worked Examples

Example 1: Fair Coin

Suppose you have a fair coin with outcomes Heads and Tails, each with probability 0.5.

• Input to the calculator: 0.5, 0.5
• Computation:
  • Each term: .
  • Sum: .
  • Apply the negative sign: bit.
• Interpretation: Each coin flip generates 1 bit of information. This is the reference level for a binary, maximally unpredictable event.

Example 2: Biased Coin

Now consider a biased coin that lands Heads with probability 0.9 and Tails with probability 0.1.

• Input to the calculator: 0.9, 0.1
• Computation (approximately):
  • .
  • .
  • Sum: .
  • Apply the negative sign: bits.
• Interpretation: This distribution is much more predictable than a fair coin, so its entropy is lower than 1 bit. You gain less information on average from observing each flip.

Example 3: Four-Outcome System

Consider a source with four possible symbols A, B, C, D that occur with probabilities 0.4, 0.3, 0.2, and 0.1 respectively.

• Input to the calculator: 0.4, 0.3, 0.2, 0.1
• Interpretive check: The probabilities sum to 1, so they form a valid discrete distribution.
• Expected magnitude: The maximum entropy for four outcomes is bits (when they are all equally likely). Because the distribution is skewed, the entropy will be less than 2 bits.

When you run this example, the calculator will show a value a bit lower than 2 bits, reflecting the unequal probabilities.

Comparison: Equal vs Skewed Distributions

The table below compares entropy values for a few simple distributions (values are approximate):

For a fixed number of outcomes, the entropy is highest when all outcomes are equally likely and decreases as the distribution becomes more imbalanced.

Why Base 2 and What About Other Bases?

In this calculator, the logarithm is base 2, so the output is in bits. This choice matches digital storage and communication systems, which fundamentally operate on binary digits.

In some applications you might see other bases:

• Natural logarithm (base e): Entropy is measured in nats, often used in statistics and physics.
• Base 10 logarithm: Entropy is measured in bans or dits, occasionally used in certain information-theoretic contexts.

To convert from bits to another base, you can multiply by an appropriate constant. For example, 1 bit is equal to nats.

Common Applications

• Data compression: The entropy gives a lower bound on the average number of bits needed per symbol when compressing a source without losing information.
• Error-correcting codes: Entropy influences how efficiently information can be transmitted over noisy channels while controlling the error rate.
• Machine learning: Classification models use entropy-related quantities (such as cross-entropy loss and information gain) to measure uncertainty and guide training.
• Cryptography: High-entropy keys and random numbers are crucial for secure encryption schemes.
• Statistics and physics: Shannon entropy is closely related to concepts of randomness, disorder, and diversity in many fields.

Limitations and Assumptions

While Shannon entropy is a powerful and widely used measure, it has important limitations and assumptions you should keep in mind when using this calculator:

• No structure beyond probabilities: Entropy depends only on the probability distribution, not on the meaning of the outcomes. Two very different systems with the same probabilities will have the same entropy.
• Requires a complete distribution: The interpretation of the result assumes that you have accounted for all possible outcomes. Omitting rare outcomes can underestimate the true entropy.
• Assumes independent, identically distributed draws: The usual interpretation of entropy as "average information per symbol" assumes repeated, independent draws from the same distribution. Correlations across time or space are not captured.
• Sensitive to input quality: If the probabilities are only rough estimates or taken from a small sample, the calculated entropy will inherit those uncertainties.
• Discrete-only: This tool does not compute differential entropy for continuous distributions, which is a related but distinct concept.
• Not a measure of complexity or value: Higher entropy means more unpredictability, but this does not always correspond to being more complex, useful, or meaningful in a practical sense.

By respecting these assumptions and limitations, you can use the calculator as a reliable tool for quantifying uncertainty and information content in a wide variety of discrete systems, from simple coins and dice to complex communication channels and machine learning models.