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Entropy (Shannon) Calculator

Average level of information/uncertainty.

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Entropy (Bits)

Formula first

Overview

Shannon entropy quantifies the average level of uncertainty, surprise, or information inherent in a random variable's possible outcomes. It provides the theoretical foundation for data compression by defining the minimum average number of bits required to represent a message.

Symbols

Variables

H = Entropy (Bits), p = Probability (p)

Entropy (Bits)
bits
Probability (p)
Variable

Apply it well

When To Use

When to use: Use this formula to determine the limits of lossless data compression or to measure the unpredictability of a discrete probability distribution. It is most effective when the set of possible outcomes is finite and their probabilities are independent and known.

Why it matters: It is the fundamental metric of information theory, enabling the efficiency of modern digital communications, from ZIP files to streaming video. By identifying the statistical structure of data, it allows for the optimization of storage and transmission bandwidth.

Avoid these traps

Common Mistakes

  • Using natural log instead of log2.
  • Forgetting both p and q terms.

One free problem

Practice Problem

A fair coin has two outcomes, heads and tails, each with a probability of 0.5. Calculate the Shannon entropy of a single coin flip.

Probability (p)0.5

Solve for:

Hint: When outcomes are equally likely (p = 0.5 for binary), entropy is at its maximum value.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Shannon, C. E. (1948). A Mathematical Theory of Communication.
  2. Cover, T. M., & Thomas, J. A. (2006). Elements of Information Theory.
  3. Wikipedia: Shannon entropy
  4. Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379-423.
  5. Cover, T. M., & Thomas, J. A. (2006). Elements of Information Theory (2nd ed.). Wiley.
  6. Claude E. Shannon, 'A Mathematical Theory of Communication', Bell System Technical Journal, 1948
  7. Thomas M. Cover and Joy A. Thomas, 'Elements of Information Theory', 2nd ed., Wiley-Interscience, 2006
  8. David J. C. MacKay, 'Information Theory, Inference, and Learning Algorithms', Cambridge University Press, 2003