What Entropy Measures in Information Theory
Entropy is a quantitative measure of uncertainty. In information theory, Shannon entropy describes how unpredictable an outcome is when it is drawn from a probability distribution. A distribution where one outcome dominates has low entropy because the result is mostly predictable. A uniform distribution has high entropy because every outcome is equally likely and uncertainty is maximized.
Entropy shows up in many areas: compression (how many bits are needed per symbol), cryptography (randomness and unpredictability), machine learning (decision tree impurity and split selection), communications (channel capacity), and statistics (uncertainty in variables). This calculator supports multiple entropy-related metrics so you can use it for coursework, analytics, and practical modeling.
Shannon Entropy Formula and Units
Shannon entropy is defined as:
H(X) = − Σ p(x) · log(p(x))
The log base determines the unit:
- Base 2: entropy is measured in bits.
- Base e: entropy is measured in nats.
- Base 10: entropy is measured in Hartleys (or bans).
This calculator lets you pick the base so results match your textbook, library, or research convention.
Entropy from Probabilities vs. Entropy from Counts
If you already have probabilities, entropy is computed directly. If you have counts (like class counts, token counts, or event counts), the calculator converts them into probabilities by dividing each count by the total. This is especially helpful in machine learning and dataset analysis, where you usually start from label counts.
If probabilities do not sum to exactly 1 due to rounding, you can enable normalization. Normalization scales all values so that the total probability equals 1, which ensures entropy calculations are valid.
Effective Number of States
A useful interpretation of entropy is the “effective number of equally-likely outcomes.” In bits, this is commonly computed as 2H. If H = 2 bits, the effective number of states is 4. If H = 0.5 bits, the effective number of states is about 1.41, meaning the distribution behaves like roughly 1.41 equally-likely outcomes.
The calculator shows this value to help interpret entropy beyond raw units. If you use a base other than 2, the same idea applies with the corresponding exponential base.
Joint Entropy, Conditional Entropy, and Mutual Information
When two variables are involved, entropy extends naturally:
- Joint entropy H(X,Y): uncertainty in the pair (X,Y).
- Conditional entropy H(X|Y): remaining uncertainty in X after knowing Y.
- Mutual information I(X;Y): how much knowing Y reduces uncertainty in X (and vice versa).
The joint/conditional tab lets you input a joint probability table and computes H(X), H(Y), H(X,Y), H(X|Y), H(Y|X), and I(X;Y). This is a common workflow in information theory, feature selection, and dependency analysis.
H(X|Y) = H(X,Y) − H(Y)
I(X;Y) = H(X) + H(Y) − H(X,Y)
Information Gain for Decision Trees
Decision trees choose splits that reduce uncertainty about the class label. Entropy is a standard impurity measure, and information gain quantifies improvement:
IG = H(parent) − Σ (wi · H(childi))
The weights wi are typically proportional to child node sizes (counts) so larger children contribute more to the final score. The Information Gain tab accepts parent and child distributions as counts or probabilities and reports parent entropy, weighted child entropy, and IG. This is practical for ML coursework and feature-split evaluation.
KL Divergence and Cross-Entropy
KL divergence compares two distributions. It measures how inefficient it would be to encode outcomes from P using a code optimized for Q. KL divergence is always non-negative and becomes 0 only when the distributions match exactly.
DKL(P‖Q) = Σ p(x) · log( p(x) / q(x) )
The calculator also reports cross-entropy H(P,Q) and entropy H(P), highlighting the identity:
H(P,Q) = H(P) + DKL(P‖Q)
Because KL divergence is sensitive to zeros in Q where P is nonzero, the tool includes an epsilon option to avoid undefined log terms. This is common in ML and numerical computing when comparing empirical distributions.
Text Entropy: Estimating Information Content from Strings
Text entropy estimates unpredictability in symbols. Character entropy looks at character frequencies (including spaces, depending on your input). Word entropy looks at tokenized words. These estimates are useful for quick analysis of randomness, repetition, compression potential, and dataset diversity.
The Text Entropy tab counts tokens and computes Shannon entropy on the resulting distribution. It also reports an “effective tokens” value so you can interpret how diverse the text is. You can export the frequency table to CSV for deeper inspection.
Limitations and Interpretation Notes
Entropy calculations depend on the quality of the underlying distribution. If probabilities are estimated from small samples, entropy may be noisy. In text entropy, tokenization choices (character vs word, case handling, punctuation) influence results. For KL divergence, ensure distributions share the same support (the same categories) and handle zero probabilities carefully.
Despite these limitations, entropy is one of the most useful tools for quantifying uncertainty and comparing distributions in a consistent mathematical way.
FAQ
Entropy Calculator – Frequently Asked Questions
Common questions about Shannon entropy, log bases, information gain, KL divergence, and text entropy.
Shannon entropy measures uncertainty or information content in a probability distribution. Higher entropy means outcomes are more unpredictable, while lower entropy means outcomes are more concentrated or predictable.
Shannon entropy is H(X)=−Σ p(x)·log(p(x)) where the log base determines units: base 2 gives bits, base e gives nats, and base 10 gives Hartleys.
Bits use log base 2, while nats use log base e. They measure the same concept with different scaling. Convert using 1 nat = 1/ln(2) bits.
Yes. Provide category counts and the calculator converts them into probabilities automatically and then computes entropy.
Entropy is used in decision trees to measure impurity and choose splits via information gain. Lower entropy after a split means the split produces purer groups.
Information gain is the reduction in entropy after splitting a dataset by a feature: IG = H(parent) − Σ (wᵢ · H(childᵢ)).
Conditional entropy H(X|Y) measures remaining uncertainty in X after you know Y. It is computed as H(X|Y)=H(X,Y)−H(Y).
KL divergence D_KL(P||Q) measures how different one distribution P is from another Q. It is not symmetric and equals 0 only when P and Q match exactly.
Yes. This calculator can estimate character entropy from input text by counting character frequencies and computing Shannon entropy of that distribution.