What is Shannon entropy?

Shannon entropy measures the average amount of 'surprise' or information contained in a random variable's outcomes, given by H(X) = −Σ p(x)·log₂ p(x). Higher entropy means more unpredictability.

What's the difference between bits and nats?

Bits use log base 2 (the standard unit in computing and information theory); nats use the natural log (base e). The same distribution has different numeric entropy values in each unit, related by a constant factor of ln(2).

What does zero entropy mean?

Zero entropy means one outcome has probability 1 and all others have probability 0 — there's no uncertainty at all, since the outcome is already certain.

What is the maximum possible entropy?

For n equally likely outcomes, entropy is maximized at log₂(n) bits, achieved exactly when every outcome has the same probability 1/n — the most unpredictable distribution possible.

Why must the probabilities sum to 1?

A probability distribution must account for every possible outcome exactly once, so the probabilities of all outcomes are required to add up to 1 (100%) for the entropy formula to be meaningful.

Where is entropy used in real life?

It underlies lossless data compression (Huffman and Shannon-Fano coding), decision tree splitting criteria in machine learning, cryptographic randomness measurement, and quantifying diversity in ecology and economics.

How is entropy related to variance?

Both measure spread or unpredictability, but variance measures numeric spread around a mean for numeric outcomes, while entropy measures unpredictability of any categorical outcome regardless of numeric value — a coin flip has entropy even though 'heads' and 'tails' aren't numbers.

Can entropy be negative?

No — since every probability is between 0 and 1, log₂(p) is always zero or negative, and the formula's leading minus sign makes every term (and the total) zero or positive.

Entropy Calculator — Compute Shannon Entropy of a Probability Distribution

Our Entropy Calculator measures the average information content — or uncertainty — of a probability distribution using Claude Shannon's entropy formula H(X) = −Σ p(x)·log₂ p(x). Enter any list of probabilities that sum to 1, and instantly see the entropy in bits (log base 2) or nats (natural log), along with the maximum possible entropy for that many outcomes.

Quick Answer

Shannon entropy H(X) = −Σ p(x)·log₂ p(x) measures the average uncertainty of a probability distribution. Enter one probability per line below (summing to 1) to instantly compute entropy in bits or nats.

Probabilities (one per line)

Enter one probability per line, e.g. 0.5. All probabilities must sum to 1.

How to Use the Entropy Calculator — Shannon Entropy (Bits & Nats) Online

1
Enter one probability per line for every possible outcome (they must sum to 1).
2
Choose bits (log₂) or nats (ln) as your unit.
3
Click 'Calculate' to see the Shannon entropy and the step-by-step breakdown.

Why Use Entropy Calculator — Shannon Entropy (Bits & Nats) Online?

Shannon entropy, introduced by Claude Shannon in his foundational 1948 paper on information theory, quantifies how unpredictable a random variable is: a distribution where one outcome is certain has zero entropy, while a uniform distribution over n outcomes has the maximum possible entropy, log₂(n) bits. This calculator applies the formula exactly so you can measure the information content of a distribution — useful for data compression, machine learning, cryptography, and thermodynamics-inspired analyses — without manually working through the logarithms.

Frequently Asked Questions

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