# calculate

Alphabet of symbols in the string:_ a e l m p s t x
Frequencies of alphabet symbols:

• 0.091 -> _
• 0.091 -> a
• 0.182 -> e
• 0.091 -> l
• 0.091 -> m
• 0.091 -> p
• 0.091 -> s
• 0.182 -> t
• 0.091 -> x

Shannon entropy can be calculated as follow:

H(X) = -[(0.091log20.091)+(0.091log20.091)+(0.182log20.182)+(0.091log20.091)+
(0.091log20.091)+(0.091log20.091)+(0.091log20.091)+(0.182log20.182)+(0.091log20.091)]

H(X) = -[(-0.314)+(-0.314)+(-0.447)+(-0.314)+(-0.314)+(-0.314)+(-0.314)+(-0.447)+(-0.314)]
H(X) = -[-3.0958]
H(X) = 3.0958

Ok, but what does it mean?
Shannon entropy tells you what is the minimal number of bits per symbol needed to encode the information in binary form (if log base is 2). Given above calculated Shannon entropy rounded up, each symbol has to be encoded by 4 bits and your need to use 44 bits to encode your string optimally.

Additionally, other formulas can be calculated, one of the simplest is metric entropy which
is Shannon entropy divided by string length. Metric entropy will help you to assess the randomness of your message. It can take values from 0 to 1, where 1 means equally distributed random string.Metric entropy for above example is: 0.28144

For further details see Wikipedia and Wikibooks pages about it.

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Date: 18:45, 23rd October 2021