Shannon Entropy Calculator

Quantify uncertainty, information content, and randomness. Compute entropy from probability distributions or any text input. Visualize probability mass and entropy decomposition — fundamental for communication theory, AI, and statistical inference.

? Fair coin (0.5,0.5) ⚖️ Biased coin (0.8,0.2) ? Fair die (1/6 six times) ? Deterministic (1,0,0) ? Max entropy 4 symbols (0.25 each)
Values are normalized to sum to 1. Non‑negative numbers only. Zero probabilities are ignored in entropy calculation.
Local computation: All entropy calculations are performed inside your browser. No data is uploaded or stored.

What is Shannon Entropy?

Claude Shannon, the father of information theory, introduced entropy in his landmark 1948 paper "A Mathematical Theory of Communication". Shannon entropy measures the average level of "information" or "uncertainty" inherent in a random variable's possible outcomes. For a discrete distribution with probabilities p₁, p₂, …, pₙ, entropy H is defined as:

H(X) = - Σ pᵢ log₂ pᵢ (bits per symbol)

Entropy reaches its maximum when all outcomes are equally probable (maximum uncertainty), and minimum (zero) when one outcome is certain. This concept revolutionized data compression (source coding theorem), cryptography, statistical inference, and even machine learning (decision trees use information gain, i.e., entropy reduction).

How the calculator works

In Probability mode, you provide weights or probabilities (e.g., "0.3,0.7" or raw counts "2,5,3"). The engine normalizes them, removes zeros, and computes H = -∑(p_i * log₂(p_i)). The chart displays the probability mass function. In Text mode, the tool counts character frequencies (including spaces, digits, punctuation), builds the empirical distribution, and computes the entropy of the message. High textual entropy indicates unpredictability, while low entropy suggests redundancy or patterns.

Case Study: Weather Prediction & Information Gain

Meteorologists use entropy to quantify forecast uncertainty. Suppose a location has three weather states: sunny (0.7), rainy (0.2), cloudy (0.1). The Shannon entropy H = 1.156 bits, indicating moderate predictability. If new data splits into morning/afternoon, the conditional entropy might drop — this information gain measures the value of additional sensors. Our calculator helps benchmark uncertainty before and after model refinements.

Properties & Applications

  • Data compression: Shannon's source coding theorem states that entropy H sets the lower bound for lossless compression.
  • Cryptography: High entropy passwords / keys resist brute‑force attacks. Randomness tests rely on entropy estimation.
  • Biodiversity indices: Shannon index (entropy) quantifies species diversity in ecology.
  • Natural Language Processing: Cross‑entropy and perplexity evaluate language models.
  • Decision trees (ID3, C4.5): Choose splits maximizing information gain = H(parent) - Σ weighted H(child).
Probability distribution p(x) Entropy (bits) Interpretation
Deterministic (1, 0, 0, ...) 0.000 No uncertainty
Fair coin (0.5, 0.5) 1.000 1 bit per toss
Biased coin (0.9, 0.1) 0.469 Less surprising on average
Fair die (6 outcomes, p=1/6 each) 2.585 ~2.585 bits/roll
English text (character entropy ~4.0‑4.5 bits) ≈4.1 (empirical) Redundancy due to language patterns

Entropy and the Second Law of Thermodynamics

Shannon's entropy shares formal similarity with Boltzmann‑Gibbs entropy in statistical mechanics, though they apply to different domains. Both quantify disorder or uncertainty. The concept has been extended to quantum information, relative entropy (Kullback‑Leibler divergence), and differential entropy for continuous variables.

This implementation follows standard information theory as taught at leading universities (MIT, Stanford). The entropy formula uses base‑2 logarithm to yield bits, consistent with Shannon's original definition. For probability normalization and zero‑handling (x log₂(0) defined as 0) we follow mathematical convention. Validation results: Fair coin → exactly 1 bit; deterministic → 0 bits; uniform 6‑symbol → log₂(6)=2.58496 bits; all tested distributions match known theoretical values (test vectors verified against NIST SP800-90B entropy assessment).

Core references: Shannon, C.E. (1948). "A Mathematical Theory of Communication"; Cover, T.M. & Thomas, J.A. (2006). "Elements of Information Theory" (Wiley); MacKay, D.J.C. (2003). "Information Theory, Inference, and Learning Algorithms" (Cambridge University Press).

Frequently Asked Questions

Log base 2 measures information in bits, the fundamental unit of digital information. One bit corresponds to a choice between two equally likely alternatives. Natural log (nats) or base 10 (hartleys) can also be used, but bits are standard in computing and communications.

By convention, 0·log₂(0) = 0, as the limit approaches zero. The calculator ignores zero probabilities automatically.

No. For a given number of symbols N, max entropy is log₂(N); our normalized entropy shows efficiency relative to that. Entropy is bounded between 0 and log₂(N).

Redundancy = 1 − (H / H_max). It quantifies the fraction of 'wasted' information or predictability. English text redundancy is around 50%.
References: Shannon, C.E. (1948). "A Mathematical Theory of Communication". Bell System Technical Journal; Cover, T.M. & Thomas, J.A. (2006). "Elements of Information Theory"; MacKay, D.J.C. (2003). "Information Theory, Inference, and Learning Algorithms".