Shannon Entropy Calculator

How much information does an outcome carry? Claude Shannon answered in 1948 with one of the most consequential formulas of the century: H = −Σ p·log₂p, the expected surprise of a distribution, measured in bits. A fair coin holds exactly 1 bit; a biased one less; a loaded die less again — and this calculator computes the figure for any set of outcomes you give it, from counts or probabilities, together with the maximum entropy the same number of outcomes could have had and the efficiency between them.

Surprise, Averaged: Reading the Formula

Shannon’s insight was to define the information in a single outcome as −log₂p — the rarer the event, the more its occurrence tells you, with the logarithm making independent information add (two coin flips carry the sum of their surprises, not the product). Entropy is then simply the average information per draw from the distribution — the expected value of surprise. Three anchor cases calibrate the intuition. A fair coin: −0.5·log₂0.5 − 0.5·log₂0.5 = 1 bit exactly, the unit’s definition. A coin biased 0.9/0.1: −(0.9·log₂0.9 + 0.1·log₂0.1) ≈ 0.469 bits — the biased coin mostly tells you what you already expected, and its information is halved for it. A certainty (one outcome at p = 1) has entropy 0: a source that always says the same thing communicates nothing, however loudly. Between those anchors lies every distribution you will ever enter, and the calculator’s maximum-entropy line marks the ceiling: for n outcomes the maximum is log₂n, reached only by the uniform distribution — which is why uniform randomness is information’s richest ore and every compression algorithm’s worst enemy.

Worked Example: Fifty, Thirty, Twenty

The defaults are counts of 50, 30, 20 — normalised to probabilities 0.5, 0.3, 0.2. Each outcome contributes −p·log₂p: for p = 0.5, that is 0.5 × 1 = 0.5000; for p = 0.3, −log₂(0.3) ≈ 1.737, times 0.3 gives 0.5211; for p = 0.2, −log₂(0.2) ≈ 2.322, times 0.2 gives 0.4644. Total H ≈ 1.4855 bits. The maximum for three outcomes is log₂3 ≈ 1.5850, so the distribution runs at 93.7% efficiency — a mere 0.1 bits of redundancy separates it from uniform. The same arithmetic governs real channels: a source alphabet used unevenly (English letters, where E carries about five times the share of an average letter) runs at visibly lower efficiency against its 26-letter ceiling, and that gap — the redundancy — is precisely what compression exploits and what prediction feeds on. Shannon’s estimate for printed English, from his famous prediction experiments, came out near 1 bit per letter against a 4.7-bit ceiling: language is mostly redundancy, which is why you can read it with letters missing.

Where the Number Earns Its Keep

Four working territories. Compression: entropy is the theoretical floor on average code length — no lossless scheme beats H bits per symbol on average, and Huffman and arithmetic coding are the engineering that approaches the floor; a file whose source entropy is its own size is incompressible in principle, not in practice. Passwords and keys: a password’s strength is its entropy — a uniform 8-character password from 64 symbols carries 48 bits; human-chosen ones carry a small fraction of that, because human choices are distributions like the 50/30/20 example, piled toward the predictable. Decision trees and machine learning: information gain — a parent entropy minus the weighted child entropies — is the splitting criterion of the classic tree algorithms; the calculator’s formula, applied twice, is the whole computation. Communications: channel capacity (Shannon’s second great theorem) caps the rate at which entropy can cross a noisy channel with vanishing error — the bits this page counts are the currency both theorems trade in, and the reason a 1948 paper about telephones now sits upstream of every digital system you own.

Units and Cousins

The unit is a dial on the same formula, set by the logarithm’s base: base 2 gives bits; base e gives nats (multiply bits by ln 2 ≈ 0.6931); base 10 gives digits or hartleys (multiply by log₁₀2 ≈ 0.3010) — the note under the answer converts your result into all three. Two cousins deserve naming so they are not mistaken for this calculator. Joint and conditional entropy extend H to pairs and contexts (the entropy of the next letter given the previous — the quantity Shannon’s English estimate really measured, and the reason pairs of letters are more compressible than singles). And thermodynamic entropy shares the mathematics and Shannon’s deliberate borrowing of the name — Boltzmann’s S = k·log W counts microstates as H counts outcomes, and the kinship between information and physics that opened with this formula now runs through black-hole thermodynamics. The calculator counts the informational original; the family resemblance is history’s compliment to it.

A Loaded Die, Weighed

One more calibration point between the coin and the letter frequencies: a die loaded so that one face lands half the time and the other five split the rest evenly — probabilities 0.5, 0.1, 0.1, 0.1, 0.1, 0.1 (enter them, or the counts 5, 1, 1, 1, 1, 1). Its entropy: the heavy face contributes 0.5 bits; each light face contributes 0.1 × 3.322 ≈ 0.332 bits, five of them 1.661 — a total of 2.161 bits against the fair die’s 2.585, an efficiency of 83.6%. The missing 0.42 bits are the price of the loading, stated in the currency casinos think in: a player who knows the distribution owns an edge measurable in exactly these bits per roll — information, once again, as the difference between the world and the uniform benchmark it deviates from.

Entropy at Play: The Guessing Game

Word games made entropy a household experience without naming it. A five-letter guessing game whose answer is drawn uniformly from a list of about 2,315 accepted answers holds log₂(2315) ≈ 11.2 bits of uncertainty at the first guess; every clue (greens, yellows, greys) is valuable exactly in proportion to the entropy it removes, and the celebrated opening words are simply the ones whose expected information gain — parent entropy minus expected child entropy, this page’s formula applied to partitions — is largest. Players who feel a guess was “wasted” on an unlikely answer are feeling entropy accounting in their fingers: the bits were the budget, and the guess spent them where the probability was not.

Related Tools

More from the Statistics Calculators hub: the Bertrand’s Box Paradox Calculator for weighing evidence the way entropy weighs outcomes, the live Benford’s Law Calculator for another distribution with surprising structure, and the AND Probability Calculator for the multiplications entropy’s logarithms turn into additions.

Frequently Asked Questions

What does an entropy of 1.4855 bits actually mean?

Three equivalent readings. As expected surprise: each draw from that distribution delivers, on average, 1.4855 bits of information. As coding floor: the outcomes can be encoded losslessly at an average of just over 1.49 bits each, and no scheme does better. As equivalent choices: the distribution carries as much uncertainty as a uniform choice among 2^1.4855 ≈ 2.8 equally likely outcomes — between a coin and a three-way choice, which the 93.7% efficiency figure says in another dialect.

Can I enter probabilities directly instead of counts?

Yes — the calculator normalises whatever you enter so the values sum to 1, so 0.5, 0.3, 0.2 and 50, 30, 20 and 5, 3, 2 all describe the same distribution and return the same entropy. Enter raw frequencies from your data, a theoretical distribution, or survey shares; normalisation handles the bookkeeping, and zero entries are gracefully ignored (an outcome that never occurs contributes no surprise).

Why is uniform the maximum?

Because any tilt toward one outcome lowers expected surprise: probability mass moved to a likelier outcome reduces its −log₂p contribution on every draw, while the starved outcome’s higher surprise arrives too rarely to compensate. The calculus (and Jensen’s inequality, for the formalists) confirms the picture at every n: H ≤ log₂n, with equality exactly at uniformity — the theorem underneath the calculator’s maximum line and underneath every “perfectly random” benchmark in cryptography.

What is the entropy of a six-sided die? A deck draw?

A fair die: log₂6 ≈ 2.585 bits. A single card drawn from a shuffled deck: log₂52 ≈ 5.700 bits. A fair coin flipped ten times, as one combined outcome: exactly 10 bits — entropy’s additivity across independent draws, and the reason key lengths are quoted in flips: a 128-bit key is a sequence as uncertain as 128 fair coins, and no fewer honest words describe what brute force must search.

Is high entropy good or bad?

Neither — it is a measurement, and its desirability flips with the application. For passwords and randomness, high entropy is the goal; for communication and storage, the source’s entropy is the cost floor and redundancy is the compressible waste; for a medical test’s outcomes, the entropy of results measures the test’s information yield before any thresholding throws most of it away. Shannon’s formula is a scale, not a verdict — this page weighs, and the weighing is yours to interpret.

How is this related to the entropy in physics?

By design and by depth: Shannon adopted the name on the advice (so the famous anecdote runs) that the formula matched Boltzmann’s, and the match is exact in structure — both are logarithms of multiplicity, one of messages, one of microstates. Modern information physics treats the two as aspects of one quantity; for this calculator’s purposes the lineage is honour enough, and the bits it returns are the same bits whether a telecom engineer or a statistical mechanician reads them.