How this instrument works
Claude Shannon's 1948 paper 'A Mathematical Theory of Communication' introduced entropy as a way to quantify uncertainty, or equivalently, the average amount of information gained when you learn the outcome of a random event. If an outcome is certain (probability 1), learning it tells you nothing new, so entropy is 0. If every outcome is equally likely, entropy is at its maximum for that number of outcomes, because you genuinely can't predict which one will occur.
This calculator uses Shannon's own original convention: entropy measured in bits, computed with log base 2. The formula is H = −Σ p_i × log2(p_i), summed over every outcome's probability p_i. A fair coin (two equally likely outcomes, each p=0.5) gives H = 1 bit exactly — one fair binary choice is, by definition, one bit of information. Four equally likely outcomes give H = 2 bits (log2(4) = 2), the maximum possible entropy for four categories; any uneven split among those four categories produces a lower entropy than that maximum.
This bits/log-base-2 framing is deliberately distinct from this site's separate shannon-index calculator, which reports the same underlying Shannon-entropy concept using natural logarithms ('nats'), framed for ecological species-diversity work rather than information theory — a convention difference, not a different formula. The numeric value of entropy always depends on which logarithm base you use (bits vs. nats vs. other units), so make sure you're comparing entropy figures computed in the same base before drawing conclusions from them.
- Enter each outcome's probability into Probabilities (must sum to 1), separated by commas, spaces, or new lines — for example 0.5, 0.5 for a fair coin, or 0.7, 0.2, 0.1 for a three-way split.
- Make sure the probabilities sum to 1 (within a small tolerance) — the calculator checks this and will flag it if they don't.
- Read Shannon entropy (bits) — a higher value means the distribution is closer to uniform (more unpredictable); a lower value means it's more concentrated on a few outcomes (more predictable).
Worked example — a fair coin, two outcomes at 0.5 each
Enter 0.5, 0.5 into Probabilities (must sum to 1). The instrument computes −(0.5 × log2(0.5) + 0.5 × log2(0.5)). Since log2(0.5) = −1, each term becomes 0.5 × −1 = −0.5, and the two terms sum to −1, which the leading minus sign flips to +1.
Shannon entropy (bits) reads 1.000000 exactly — Shannon's own textbook baseline case. One fair binary choice, like a coin flip, carries exactly 1 bit of information by definition; that's precisely why the unit is called a 'bit' (binary digit) in the first place. A biased coin — say 0.9 and 0.1 — would produce noticeably less than 1 bit, since a lopsided outcome is easier to guess in advance and therefore carries less genuine uncertainty.
Questions
What does a Shannon entropy of 1 bit actually mean?
It means the distribution carries exactly as much uncertainty as one fair coin flip — the theoretical baseline unit. Concretely, 1 bit of entropy is the amount you'd need, on average, to correctly encode which outcome occurred using an optimal binary (yes/no) encoding scheme. Two equally likely outcomes always produce exactly 1 bit; anything more predictable than that produces less than 1 bit, and more categories at higher uncertainty can produce more.
Why is this different from the shannon-index calculator on this site?
Both compute the same underlying Shannon-entropy formula, but with a different logarithm base and a different framing. This calculator uses log base 2 and reports the result in 'bits,' Shannon's own original information-theory convention. The site's separate shannon-index calculator uses the natural logarithm ('nats'), the convention typically used in ecology for measuring species diversity. The two aren't competing formulas — they're the same math reported in two different, field-specific units, which is why the numeric values differ even for the same input probabilities.
What's the maximum possible entropy for a given number of outcomes?
The maximum is log2(n) bits for n equally likely outcomes, achieved exactly when every outcome has the same probability. Two outcomes max out at 1 bit; four outcomes max out at 2 bits; eight at 3 bits, and so on. Any deviation from a perfectly uniform distribution over those n outcomes will produce a strictly lower entropy than that maximum.
Can Shannon entropy be zero?
Yes — entropy is exactly zero when one outcome has probability 1 (certainty) and all others have probability 0. There's no uncertainty left to measure: you already know exactly what will happen, so learning the outcome provides zero new information.
Why must the probabilities sum to 1?
Because they're meant to represent a complete probability distribution over all possible outcomes of the event — every outcome that could happen has to be accounted for, and the total probability across all of them is, by definition, 1 (or 100%). If your probabilities don't sum to 1, they aren't a valid probability distribution yet, and the entropy formula's result won't be meaningful.