Shannon Entropy Calculator

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Shannon Entropy Calculator

Measure information content and uncertainty in data using Shannon's information theory.

Last updated: March 2026

Calculator

Shannon Entropy (H)
1.8829
Bits
Max Entropy2.0000
Redundancy5.9%
Unique Symbols4
Total Symbols16

Symbol Distribution

A
37.5% (6)
B
31.3% (5)
C
18.8% (3)
D
12.5% (2)

What is Shannon Entropy?

Shannon entropy is a measure of the average information content or uncertainty in data. Introduced by Claude Shannon in 1948, it quantifies how much randomness or diversity is present—high entropy means high uncertainty (many equally likely outcomes), low entropy means predictability (few likely outcomes). Entropy is fundamental to information theory, data compression, cryptography, and machine learning.

Entropically, a fair coin flip (50/50 heads/tails) has maximum entropy compared to a biased coin (90/10). A uniform distribution of symbols has higher entropy than one where symbols are unevenly distributed. Entropy is measured in different units depending on the logarithm base: bits (base 2), nats (base e), or hartleys (base 10).

The formula H = −Σ(pᵢ × log(pᵢ)) calculates average information. Redundancy measures how far actual entropy is from maximum entropy—high redundancy means the data is predictable and compressible.

How to Calculate Shannon Entropy

Step-by-Step Process

Step 1: List your symbols/values (text, numbers, categories)
Step 2: Enter them separated by commas or spaces
Step 3: Choose logarithm base (2 for bits is most common)
Step 4: View entropy, frequencies, and distribution

Key Formulas

Shannon Entropy:
H = −Σ(pᵢ × logᵦ(pᵢ))
Maximum Entropy:
H_max = logᵦ(n) where n = number of unique symbols
Redundancy:
R = 1 − (H / H_max)
Units by Base:
Base 2 = bits, Base e = nats, Base 10 = hartleys

Real-World Example

Text Analysis: Which is More Unpredictable?

Text A:
"AAAAAABBBBCC" (5 As, 4 Bs, 2 Cs)
Unevenly distributed, predictable
Text B:
"ABCABCABCABC" (4 As, 4 Bs, 4 Cs)
Evenly distributed, less predictable
Results:
Text A: H ≈ 1.52 bits
Lower entropy due to uneven distribution. More compressible.
Text B: H ≈ 1.58 bits
Higher entropy due to uniform distribution. Less compressible.

Frequently Asked Questions

What does negative entropy mean?

Entropy is always non-negative (≥ 0). A negative result indicates a calculation error. Entropy = 0 only when one symbol occurs with certainty.

Why different log bases?

Base 2 (bits) is common in computer science. Base e (nats) appears in physics. Base 10 (hartleys) in telecommunications. They're just different units—multiply by log(b)/log(b') to convert.

What's the difference between entropy and redundancy?

Entropy measures actual information content. Redundancy measures unused potential—how much less random the data is than maximum. Low redundancy (near 1) means data is nearly random.

How is Shannon entropy used in practice?

Data compression (ZIP uses entropy principles), cryptography (strong encryption maximizes entropy), machine learning (information gain measures feature importance), quality control (detecting process changes).

Can entropy values exceed the maximum?

No, entropy is always ≤ log₂(n) where n is unique symbols. If you get H > max H, check your calculation. Maximum entropy occurs with uniform probability distribution.

Is higher entropy always better?

It depends! For cryptographic keys, higher entropy is better (harder to crack). For data compression, lower entropy is better (more compressible). Context matters.

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