Information theory formalizes the concept of information as a measure of surprise. When an event is certain, observing it conveys no new information. Conversely, rare events carry more information because they reduce uncertainty by a greater amount. Claude Shannon introduced a mathematical expression for this notion in 1948, giving rise to the field of digital communication. His entropy formula captures how unpredictable a source is and sets fundamental limits on data compression and channel capacity.
For a discrete random variable with outcomes occurring with probabilities , Shannon entropy is defined as . The logarithm base 2 means entropy is measured in bits. If all outcomes are equally likely, entropy reaches its maximum because each observation tells you as much as possible.
Using a base-2 logarithm relates entropy directly to binary information. One bit corresponds to a choice between two equally likely alternatives. If an event has probability one-half, it contributes one bit of entropy. You can, however, express entropy in other units by choosing a different base. For example, using the natural logarithm yields units of nats, common in statistical mechanics.
Consider a fair coin flip. With probabilities [0.5, 0.5], the entropy is exactly one bit. A biased coin with probabilities [0.9, 0.1] has lower entropy, about 0.47 bits, because the outcome is more predictable. In data compression, this means shorter codewords can represent the more probable outcome. For larger alphabets, such as letters in English text, entropy quantifies average information per character and guides the design of efficient coding schemes.
An entropy of zero implies complete certainty: one of the probabilities is one and the rest are zero. As probabilities become more evenly distributed, entropy increases, reflecting greater uncertainty. The maximum entropy for equally likely outcomes is bits. This sets an upper bound on how much you can compress data without losing information. Random sources with high entropy resist compression, while structured sources with low entropy can be represented more concisely.
Entropy underpins a wide range of modern technology. It informs the design of error-correcting codes that protect digital transmissions against noise. In cryptography, random key generation relies on high-entropy sources to ensure security. Machine learning models use entropy-based loss functions to measure prediction uncertainty. Even thermodynamics connects to information theory through the concept of statistical entropy, linking microscopic states to macroscopic disorder.
To compute entropy, enter a list of probabilities separated by commas. They should sum to one, although the calculator will normalize them if they do not. Each probability must be positive. The script converts the input into an array, divides each value by the total to obtain normalized probabilities, and applies Shannon’s formula. The resulting entropy appears below the form in bits.
Suppose you are analyzing a simple four-symbol code with probabilities [0.1, 0.2, 0.3, 0.4]. Plugging these into the calculator yields about 1.85 bits of entropy. That means, on average, each symbol conveys less than two bits of information, so any lossless encoding scheme must use at least this many bits per symbol. Real-world communication systems often approach this theoretical limit but never go below it.
If you vary one probability while keeping the others fixed, you will notice how the entropy changes smoothly. Small shifts in probability near zero or one have little effect because those outcomes already carry minimal uncertainty. The curve is steepest when probabilities are near equal, reflecting how a balanced choice offers the most information. By experimenting with different values, you can build intuition for how randomness translates to information content.
In machine learning, entropy helps evaluate how well an attribute splits a dataset. Decision tree algorithms choose the attribute that reduces entropy the most, leading to more uniform subsets and better classification performance. This process, known as information gain, directly applies Shannon’s measure to real-world data problems.
While entropy quantifies average uncertainty, it doesn’t capture all aspects of a distribution. Two different distributions can share the same entropy yet behave very differently. Additionally, the formula assumes independent events; correlations between outcomes may require more sophisticated measures like mutual information. Still, Shannon entropy remains a cornerstone of information theory due to its simplicity and broad applicability.
Shannon developed his theory to address problems in telecommunications, particularly how to encode messages efficiently and transmit them reliably over noisy channels. His work drew inspiration from statistical mechanics and has since influenced fields as diverse as linguistics, neurobiology, and quantum computing. The entropy formula you calculate here continues to shape research into how systems store and convey information.
The Shannon Entropy Calculator provides an accessible way to explore how probability distributions relate to information. By experimenting with different sets of probabilities, you can see firsthand how uncertainty scales with predictability. Whether you’re designing a coding scheme, analyzing data, or just curious about the mathematics of information, this tool illustrates one of the most profound connections between probability and knowledge.
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