Entropy Calculator

Shannon entropy, introduced by Claude Shannon in 1948, measures the average uncertainty or unpredictability in a set of outcomes. It quantifies how much information is contained in a message or data source. Higher entropy means more randomness and more information content — it is widely used in data compression, cryptography, machine learning, and communication theory.

Shannon entropy is calculated using the formula H(X) = −∑ P(xᵢ) · log_b P(xᵢ), where P(xᵢ) is the probability of each outcome and b is the logarithm base. Enter your event probabilities (they must sum to 1) and choose a log base — base 2 gives entropy in bits, base e gives nats, and base 10 gives dits (decimal digits).

Entropy is maximised when all outcomes are equally probable. For n events, the maximum entropy is log₂(n) bits. This means the system is completely unpredictable — knowing nothing about the process gives you the least possible advantage in guessing the next outcome.

For a reversible process, entropy change is ΔS = Q / T, where Q is the heat transferred (in joules) and T is the absolute temperature (in Kelvin). A positive ΔS indicates the system gained entropy (disorder increased), while a negative ΔS means entropy decreased in the system — though total entropy of the universe always increases or stays the same.

Normalized entropy divides the calculated Shannon entropy by the maximum possible entropy for the same number of events, giving a value between 0 and 1. A value of 1 means the distribution is perfectly uniform (maximum disorder), while a value close to 0 means one outcome is highly dominant (very predictable).

Probabilities represent the likelihood of all possible, mutually exclusive outcomes of a random variable. Because one of those outcomes must always occur, their total must equal exactly 1 (or 100%). If your probabilities don't sum to 1, the entropy value will not be mathematically valid — this calculator warns you when the sum deviates from 1.

These are simply different units of information entropy based on the logarithm base used. Bits (base 2) are the most common in computing and information theory. Nats (base e, natural logarithm) are used in statistical mechanics and physics. Dits or hartleys (base 10) are used occasionally in communications. The underlying measurement is the same — only the scale differs.

In everyday terms, entropy describes how disordered or chaotic a system is. A tidy room has low entropy; a messy one has high entropy. In thermodynamics, it explains why heat flows from hot to cold and why perpetual motion machines are impossible. In information theory, it explains why compressed files can't be compressed again without loss — there's no remaining redundancy to exploit.