Shannon’s entropy quantifies the quantity of data during a variable, thus providing the inspiration for a theory around the notion of information.
$$ If\ we\ have\ a\ symbol\ set\ {A,B,C,D,E} where\ the\ symbol\ occurance\ frequencies\ are:$$ $$ A=0.5$$ $$ B=0.2$$ $$ C=0.1$$ $$ D=0.1$$ $$ E=0.1$$
$$ The\ average\ minimum\ number\ of\ bits\ needed\ to\ represent\ a\ symbol\ is$$ $$H(X) = -[(0.5log20.5 + 0.2log20.2 + (0.1log20.1)*3)]$$ $$H(X) = -[-0.5 + (-0.46438) + (-0.9965)]$$ $$H(X) = -[-1.9]$$ $$H(X) = 1.9$$
Shannon’s entropy quantifies the quantity of data during a variable, thus providing the inspiration for a theory around the notion of information.
Storage and transmission of data can intuitively be expected to be tied to the quantity of data involved. for instance, information may be about the result of a coin toss. This information is often stored during a Boolean variable which will combat the values 0 or 1. We will use the variable to represent the data like the coin toss, viz.,
Whether the coin toss came up heads or not. In digital storage and transmission technology, this Boolean variable is often represented during a single "bit", the essential unit of digital information storage/transmission.
However, this bit directly stores the worth of the variable, i.e. the raw data like the result of the coin toss. It doesn't succinctly capture the knowledge within the coin toss, e.g., whether the coin is biased or unbiased, and, if biased, how biased.
Whereas, Shannon’s entropy metric quantifies, among other things, the absolute minimum amount of storage and transmission needed for succinctly capturing any information (as against raw data), and in typical cases that amount is a smaller amount than what's required to store or transmit the raw data behind the knowledge. Shannon’s Entropy metric also suggests away of representing the knowledge within the calculated fewer number of bits.
The formula of Shannon Entropy
So here is the formula for calculating the Shannon entropy.
Shannon Entropy E = -∑i(p(i)×log2(p(i)))
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