Calculate Entropy
Use decimal probabilities, such as 0.25. Add or remove outcome fields with the count control.
Example Data Table
| Distribution | Base | Entropy | Meaning |
|---|---|---|---|
| 0.25, 0.25, 0.25, 0.25 | 2 | 2 bits | Four equally likely outcomes. |
| 0.50, 0.25, 0.125, 0.125 | 2 | 1.75 bits | Moderately uneven outcome chances. |
| 0.70, 0.10, 0.10, 0.10 | 2 | 1.3568 bits | One outcome strongly dominates. |
Formula Used
H is entropy. Each pi is an outcome probability. The base b selects bits, nats, or hartleys. A zero probability contributes zero by limit.
Related Measures
Maximum entropy: Hmax = logb(n), where n is the number of listed outcomes.
Evenness: (H ÷ Hmax) × 100. It compares the result with the most balanced distribution.
Effective outcomes: bH. It gives the equivalent count of equally likely outcomes.
How to Use This Calculator
- Set the number of possible outcomes.
- Enter one decimal probability for every outcome.
- Select bits, nats, or hartleys as the entropy unit.
- Choose strict validation for exact models.
- Choose auto-normalization only for small rounding differences.
- Press Calculate entropy and inspect the result panel.
- Review each contribution, then export CSV or PDF when needed.
Entropy and Probability Distributions
Entropy measures uncertainty across a set of possible outcomes. It does not measure thermal energy. A distribution with one certain outcome has zero entropy. A distribution with evenly likely outcomes has greater entropy. The logarithm base selects the unit. Base two gives bits. Natural logarithms give nats. Base ten gives hartleys. Entropy helps compare systems with different probability patterns.
Each probability describes how often an outcome occurs. All probabilities must be zero or positive. Their total should equal one. A value of zero is allowed. It adds no direct entropy contribution. A value near one dominates the distribution. That dominance lowers uncertainty. Equal probabilities spread uncertainty across the available outcomes. The calculator checks the total before reporting a result. It can normalize a positive total when values are rounded.
Why Logarithms Appear
The Shannon expression combines every nonzero probability. Small probabilities can carry substantial surprise when they occur. Their entropy contribution depends on both likelihood and surprisal. A probability of one half contributes one half bit. Two equal outcomes therefore create one bit of entropy. Four equal outcomes create two bits. Eight equal outcomes create three bits. This pattern explains why logarithms appear in formula. They connect outcome counts with information units.
Maximum and Relative Entropy
Maximum entropy depends on the number of listed outcomes. It occurs only when every listed probability is equal. The maximum is log base b of the outcome count. Comparing actual entropy with this maximum gives an evenness percentage. A high percentage means probabilities are fairly balanced. A low percentage means a few outcomes dominate. This comparison helps when models use different state counts. It separates randomness from simple outcome count.
Effective Outcome Count
Effective outcomes provide another intuitive result. The calculator raises the chosen base to the entropy value. This converts entropy into an equivalent number of equally likely outcomes. For example, two bits correspond to four effective outcomes. The value can be fractional for uneven distributions. It does not mean there is a fractional state. It summarizes how broadly probability is spread. This measure is often called perplexity in information theory.
Reliable Inputs Matter
Use entropy carefully in physical interpretation. The input probabilities must describe a clear sample space. Do not mix unrelated outcomes. Do not use percentages unless you divide by one hundred. Keep rounding consistent. Use strict mode when an exact probability model is required. Use normalization only when small rounding differences are expected. Normalization changes every probability by the same scale factor. It cannot repair a missing outcome or an incorrect model.
Interpret Results in Context
Entropy does not identify causes. It only summarizes the distribution you provide. A high value does not automatically mean a system is better or more chaotic. Context matters. In communication, entropy indicates average information. In statistical physics, probabilities may describe microscopic states. In data science, entropy can measure class uncertainty. The formula remains the same, but interpretation changes with the model. Record the unit, base, and probability source with every result.
Frequently Asked Questions
It measures uncertainty in a probability distribution. It reports the average information associated with observing an outcome. This calculator uses the Shannon entropy formula.
Yes, a complete probability distribution totals one. Strict mode checks this directly. Auto-normalization rescales a positive total, which is useful only when minor rounding created the difference.
Yes. A zero-probability outcome adds zero to entropy. The calculator handles it safely without attempting to evaluate the undefined ordinary logarithm of zero.
No. Entropy depends on the probability values, not their order. Rearranging the same probabilities leaves the total entropy unchanged.
They use different logarithm bases. Bits use base two, nats use the natural base, and hartleys use base ten. They express the same uncertainty in different units.
It is the largest possible entropy for the listed outcome count. It occurs when every outcome is equally likely. The calculator shows it in the selected unit.
Effective outcomes convert entropy into an equivalent number of equally likely states. They make an uneven distribution easier to compare with a perfectly balanced one.
Convert each percentage to a decimal first. For example, enter 25% as 0.25. The calculator expects probability values from zero through one.
No. It only rescales values by one common factor. It cannot correct missing outcomes, wrong probabilities, or an unclear sample space.
The mathematical form is closely related, but this tool calculates Shannon entropy from supplied probabilities. Thermodynamic use requires physical constants and a defined physical model.
Not always. It describes uncertainty in the selected outcomes. Physical interpretations depend on the model and constraints. Careful probability inputs produce meaningful entropy estimates every time.