Shannon Entropy Calculator

Shannon entropy is a mathematical measure of uncertainty, surprise, or average information in a probability distribution. Claude Shannon introduced it while building the foundations of information theory. For outcomes with probabilities p, entropy is calculated as the negative sum of p times log base 2 of p. The base-2 logarithm expresses the result in bits, making the value easy to connect with binary encoding and digital communication. A certain event has zero entropy because observing it provides no surprise. A fair coin has one bit of entropy because either of two equally likely outcomes can occur. Four equally likely outcomes have two bits, and eight equally likely outcomes have three bits. For a fixed number of possible symbols, entropy reaches its maximum when every symbol is equally probable. As a distribution becomes concentrated around a few likely outcomes, entropy falls because observations become easier to predict. The probability mode accepts positive decimal probabilities separated by commas, spaces, or semicolons. They must add to one, apart from tiny floating-point rounding differences. The text mode counts every character in the supplied message, converts each character frequency into a probability, and applies the same entropy formula. Spaces and punctuation are treated as symbols because they carry information too. Character-level entropy is useful for quick comparisons, though it does not account for dependencies between neighboring characters. Entropy appears in data compression, coding theory, decision trees, cryptography, ecology, language analysis, and machine learning. A compressor can represent frequent symbols with shorter codes, and entropy establishes a theoretical lower bound on the average bits needed when symbols are independent. In classification, information gain compares entropy before and after splitting data. In security analysis, high observed entropy may indicate randomness, but entropy alone cannot prove that a password, key, or generator is secure. Interpret results relative to the number of possible outcomes and the way symbols were defined. Comparing raw entropy across alphabets of different sizes can be misleading, so the displayed maximum offers useful context. Small samples can also produce unstable frequency estimates. Use representative data, preserve meaningful symbols, and combine entropy with domain knowledge when evaluating uncertainty. This calculator provides a transparent, deterministic result for education and exploratory analysis.