Matrix Calculator Desmos

Expert Verified by: David Chen, Data Scientist & Information Theory Researcher

This professional Shannon Entropy Calculator allows you to measure the uncertainty or average information content in a discrete random variable. Simply input your probabilities or frequency counts below to quantify the “surprise” factor of your data set.

Shannon Entropy Calculator

Note: If inputs don’t sum to 1, they will be treated as counts and normalized.

0.0000 Bits

Total Outcomes: 0

Shannon Entropy Formula:

$$H(X) = -\sum_{i=1}^{n} P(x_i) \log_2 P(x_i)$$

Source: A Mathematical Theory of Communication (C.E. Shannon)

Variables:

• H(X): The Shannon Entropy in bits.
• P(x_i): The probability of the i-th outcome.
• n: The total number of possible outcomes.
• log₂: Base-2 logarithm (standard for information measurement).

Related Calculators:

• Binary Entropy Calculator
• Relative Entropy (KL Divergence) Tool
• Gini Impurity Index Calculator
• Mutual Information Score Estimator

What is Shannon Entropy?

Shannon Entropy is a fundamental concept in information theory introduced by Claude Shannon in 1948. It quantifies the level of uncertainty, randomness, or complexity in a set of data. In simpler terms, it measures how much “information” is produced on average by a source of data.

A high entropy value indicates high uncertainty (e.g., a fair coin flip), whereas an entropy of zero means the outcome is certain (e.g., a two-headed coin). It is widely used today in data compression, cryptography, and machine learning (specifically in Decision Trees).

How to Calculate Shannon Entropy (Example):

1. Identify Probabilities: Suppose you have three symbols with probabilities: 0.5, 0.25, and 0.25.
2. Apply Logarithms: Calculate $\log_2$ for each: $\log_2(0.5) = -1$, $\log_2(0.25) = -2$.
3. Multiply: $P(x) \times \log_2(P(x))$. Results: $-0.5$, $-0.5$, $-0.5$.
4. Sum & Negate: Sum is $-1.5$. Negative of $-1.5$ is 1.5 Bits.

Frequently Asked Questions (FAQ):

What is the unit of Shannon Entropy? The most common unit is the “bit” (using base-2 logs), but “nats” (base-e) or “bans” (base-10) are also used.

Can Shannon entropy be negative? No. Since probabilities are between 0 and 1, the logarithm is always non-positive, and the final negation makes the entropy zero or positive.

What is maximum entropy? For a discrete variable, maximum entropy occurs when all outcomes are equally likely (Uniform Distribution).

How does it relate to Data Compression? Shannon’s Source Coding Theorem states that the entropy is the absolute mathematical limit on how much data can be compressed without losing information.