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# Shannon Entropy

##### This online calculator computes Shannon entropy for a given event probability table and for a given message
Timur2013-06-04 15:04:43

In information theory, entropy is a measure of the uncertainty in a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message.
The formula for entropy was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication".

$H(X)= - \sum_{i=1}^np(x_i)\log_b p(x_i)$

Minus is used because for values less than 1 logarithm is negative, however, since

$-\log a = \log \frac{1}{a}$,

formula can be expressed as

$H(X)= \sum_{i=1}^np(x_i)\log_b \frac{1}{p(x_i)}$

Expression
$\log_b \frac{1}{p(x_i)}$
is also called an uncertainty or surprisal, the lower the probability $p(x_i)$, i.e. $p(x_i)$ → 0, the higher the uncertainty or the surprise, i.e. $u_i$ → ∞, for the outcome $x_i$.

Formula, in this case, expresses the mathematical expectation of uncertainty and that is why information entropy and information uncertainty can be used interchangeably.

There are two calculators below - one computes Shannon entropy for given probabilities of events, second computes Shannon entropy for given symbol frequencies for given message.

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##### Shannon Entropy

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