What does this calculator do?

Compute statistical results from your inputs.

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What are the formulas?

See the educational content section for formulas.

For statistical analysis and computation.

Check the related calculators section below.

Results follow standard statistical methods.

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Shannon Entropy Calculator

Free Shannon entropy calculator. Entropy, mutual information, conditional entropy, perplexity. From

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Why This Statistical Analysis Matters

Why: Statistical calculator for analysis.

How: Enter inputs and compute results.

STATISTICSInformation Theory

Shannon Entropy — H, Joint, Conditional, Mutual Information

H(X) = −Σ pᵢ log(pᵢ). Joint entropy, conditional entropy, mutual information, perplexity. From probabilities, frequencies, or joint distributions.

Simpson's Diversity →Expected Value →

Real-World Scenarios — Click to Load

Mode

Log base

Categories & probabilities

shannon_entropy_results.sh

CALCULATED

$ shannon_entropy --mode="probabilities" --base=2

H(X)

1.0000 bits

H_max

1.0000

H_norm

1.0000

Perplexity

2.00

Shannon Entropy Result

H(X) = −Σ pᵢ log(pᵢ)

H = 1.000 bits

Perplexity: 2.00H_norm: 1.000

numbervibe.com/calculators/statistics/shannon-entropy-calculator

Entropy Contribution by Category

Probability Distribution

Entropy vs Distribution Shape

Calculation Breakdown

COMPUTATION

Shannon Entropy H(X)

1.0000 bits

H = -\text{Sigma} pᵢ \text{log}(pᵢ)

Maximum Entropy H_max

1.0000

H_max = \text{log}(k)

Normalized Entropy

1.0000

H_norm = H / H_max

Redundancy

0.0000

R = 1 - H_norm

Perplexity

2.00

2^H

For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

Shannon entropy: H(X) = −Σ pᵢ log(pᵢ). Measures uncertainty in bits (base 2), nats (base e), or hartleys (base 10).
Maximum entropy: H_max = log(k) for k categories. Uniform distribution achieves maximum.
Joint entropy: H(X,Y) = −Σᵢ Σⱼ p(xᵢ,yⱼ) log(p(xᵢ,yⱼ)).
Conditional entropy: H(Y|X) = H(X,Y) − H(X).
Mutual information: I(X;Y) = H(X) + H(Y) − H(X,Y). Measures shared information.
Perplexity: 2^H (base 2) = effective number of equally likely outcomes.

Did You Know?

🪙A fair coin has entropy 1 bit — the maximum for 2 outcomes. A biased coin has lower entropy.Source: Shannon 1948

📧English text has ~4 bits per character entropy due to letter frequency and correlations.Source: Cover & Thomas

🧬DNA with equal base frequencies has 2 bits per base. Real DNA has slightly less due to bias.Source: Genomics

🌐Network traffic entropy helps detect anomalies — uniform = normal, skewed = attack.Source: NIST

🖼️Image compression exploits redundancy. Lower entropy = more compressible.Source: JPEG/PNG

🔒Password strength ∝ entropy. 8 chars from 95 symbols ≈ 52 bits.Source: NIST SP 800-63

Formulas Reference

H(X) = −Σ pᵢ log₂(pᵢ)

Shannon entropy (bits)

H_max = log₂(k)

Maximum entropy for k categories

H(Y|X) = H(X,Y) − H(X)

Conditional entropy

I(X;Y) = H(X) + H(Y) − H(X,Y)

Mutual information

Perplexity = 2^H

Effective number of outcomes (base 2)

Choosing the Right Mode

From probabilities: Enter probabilities that sum to 1. From frequencies: Enter counts; they will be normalized. Joint/conditional: Enter a joint distribution table for two variables to compute H(X,Y), H(Y|X), and I(X;Y).

Frequently Asked Questions

What is the difference between bits, nats, and hartleys?

Bits use log₂ (information theory). Nats use ln (physics). Hartleys use log₁₀. Conversion: 1 nat ≈ 1.44 bits.

When is entropy maximized?

For a fixed number of categories, entropy is maximized when all probabilities are equal (uniform distribution).

What does mutual information measure?

I(X;Y) measures how much knowing X reduces uncertainty about Y (and vice versa). Zero if independent.

How is entropy used in data compression?

Shannon proved that entropy is the theoretical lower bound for lossless compression. Better compression = lower entropy.

What is perplexity?

Perplexity = 2^H is the effective number of equally likely outcomes. Used in language model evaluation.

How does joint entropy relate to marginals?

H(X,Y) ≤ H(X) + H(Y). Equality iff X and Y independent. Chain rule: H(X,Y) = H(X) + H(Y|X).

Applications

Data Compression

Entropy bounds lossless compression. Huffman coding approaches this limit.

Feature Selection

Mutual information identifies features most predictive of the target.

Password Strength

Character set entropy determines theoretical password space.

Language Models

Perplexity evaluates language models. Cross-entropy for training.

Worked Example

Fair coin: p = [0.5, 0.5]. H = −0.5×log₂(0.5) − 0.5×log₂(0.5) = 1 bit. Biased coin (0.8, 0.2): H ≈ 0.722 bits. Perplexity = 2^0.722 ≈ 1.65.

Chain Rule & KL Divergence

Chain rule: H(X₁,…,Xₙ) = H(X₁) + H(X₂|X₁) + … + H(Xₙ|X₁,…,Xₙ₋₁). KL(P‖Q) = Σ pᵢ log(pᵢ/qᵢ) measures how much P differs from Q. Cross-entropy H(p,q) = −Σ pᵢ log(qᵢ) is the standard loss for classification.