Shannon Entropy Calculator - Measure Information in Bits

A Shannon entropy calculator measures how unpredictable a random source is by converting its outcome probabilities into a single number called entropy. It answers the practical question 'how much information is in this distribution?' by averaging the surprise of each outcome.

• • Data compression: Estimate the smallest average code length a source needs, because entropy sets the theoretical limit that any lossless compressor can reach.
• • Fairness and bias checks: Compare a loaded die or biased coin against a perfectly fair one to see how much predictability the imbalance introduces.
• • Machine learning features: Score how much signal a categorical feature carries before feeding it into a model, since low-entropy columns carry little discriminative power.
• • Genetics and ecology: Quantify diversity in allele frequencies or species counts, where Shannon's index is the same entropy formula applied to relative abundances.

Entropy sits between 0 and log_b(N). A value of 0 means the outcome is already known, while the maximum means every outcome is equally likely and nothing about the next draw can be predicted.

The intuition that makes the Shannon entropy calculator useful is that rare events carry more information than common ones. Learning that a 1-in-1000 event happened tells you far more than learning a 99-percent event happened, and the log term is what captures that asymmetry.

Before you can measure uncertainty you need the outcome probabilities themselves, which the Probability Calculator produces from event counts or odds.

The Shannon entropy calculator applies the definition H = -SUM_i [ p_i * log_b(p_i) ] to the probabilities you supply, then reports the result in the unit chosen by your logarithm base.

• p(i): Probability of outcome i, a number between 0 and 1; all outcomes together must sum to 1.
• b: Logarithm base: 2 for bits, e for nats, 10 for dits.
• H: Resulting entropy, the average information per draw from the distribution.

For any outcome with probability exactly 0 the term p·log(p) is treated as 0, since that multiplies a vanishing probability by a diverging log and the product is defined to be 0 by continuity.

Switching the base only rescales the answer. One nat equals one bit divided by ln(2) (about 0.693), so the same distribution reported in nats is a fixed multiple of its bit value.

According to Wikipedia — Entropy (information theory), Shannon entropy H(X) = -SUM p(x) log p(x) quantifies the average information content or unpredictability of a random variable X.

The classic fair-versus-biased coin example is easiest to set up with the Coin Flip Probability Calculator, which returns the exact probabilities to paste into this tool.

Four ideas anchor the output of the Shannon entropy calculator and let you interpret the number instead of just reading it.

These concepts connect entropy to other summaries of a distribution. Where a variance measures spread on a number line, entropy measures spread across categories without assuming any ordering among them.

Where entropy weights outcomes by log-probability, the Expected Value Calculator weights them by their payoff, so the two ideas make a useful side-by-side.

Five short steps take you from a list of probabilities to an interpreted entropy value.

1. 1 List the outcome probabilities: Write one probability per line in the box, each between 0 and 1, and confirm they add up to 1.
2. 2 Choose a logarithm base: Pick base 2 for bits (default), base e for nats, or base 10 for dits depending on the convention your field uses.
3. 3 Decide on normalization: Set normalized entropy to Yes if you want a 0-to-1 value that is comparable across distributions with different outcome counts.
4. 4 Read the result panel: The panel shows raw entropy, normalized entropy, and the unit so you can quote the number directly in a report or assignment.
5. 5 Check the sum warning: If the rows do not sum to 1 the calculator refuses to run and tells you the current sum, which catches transcription errors before they skew the result.

Example: paste 0.4, 0.3, 0.2, 0.1 for a four-outcome source, leave base 2, and the calculator returns about 1.846 bits, below the 2-bit maximum of a fair four-way choice because the probabilities are uneven.

If your raw data are observations rather than probabilities, summarize them first with the Descriptive Statistics Calculator before converting counts to a distribution.

The payoff is removing hand-calculation error and making the interpretation visible at a glance.

• • Avoids log arithmetic mistakes: Summing -p·log(p) by hand across many outcomes is error-prone; the calculator does it exactly and without rounding slips.
• • Makes the fairness ceiling obvious: Showing both raw and normalized entropy lets you see how far a distribution sits below its uniform maximum.
• • Supports teaching: Students can change one probability and watch entropy move, building intuition faster than a static textbook table.
• • Unit flexibility: Switching between bits, nats, and dits in one click prevents the common mix-up of reporting nats where bits were expected.
• • Feeds downstream analysis: The entropy value drops straight into compression budgets, feature-selection scores, or diversity indexes without re-deriving it.

The biggest practical gain is speed: a compression engineer comparing ten candidate sources can rank them by entropy in seconds rather than recomputing each by hand.

For repeated trials with a fixed success chance, the Binomial Distribution Calculator gives the outcome probabilities whose entropy you can then inspect here.

Three factors decide the entropy number, and two limitations keep the result honest.

• • Entropy assumes the probabilities you enter are correct; it does not estimate them from raw data, so a wrong input produces a confidently wrong number.
• • For continuous variables this discrete formula does not apply directly; continuous entropy uses a different (differential) definition and can even be negative.

Treat the output as a property of the distribution you specified, not of reality. If your probabilities come from a small sample, the true entropy may differ until the sample is large enough.

According to C. E. Shannon, A Mathematical Theory of Communication (Harvard mirror of the 1948 Bell System paper), The entropy of a source measures the minimum average number of binary digits needed per symbol to encode its output without loss.