About the Shannon Entropy
The Shannon entropy calculator measures how unpredictable or diverse a set of categorical counts is, turning raw frequencies into a single number expressed in bits. It is the same measure used in information theory to describe how much uncertainty exists before an outcome is observed. Enter counts for each category, such as votes per candidate or specimens per species, to see how concentrated or spread out the distribution really is.
How It Works
You enter a count for each category, separated by commas. The calculator divides each count by the total to get a proportion for that category, then sums each proportion multiplied by its own base-2 logarithm, with the sign flipped, to produce the entropy value in bits. It also reports the maximum entropy possible for that many categories and a normalized 0-to-1 evenness score.
Formula & Methodology
To compute by hand, add up all the category counts to get the total, then divide each individual count by that total to get proportions p_i. For every category with a nonzero count, multiply p_i by log base 2 of p_i, then sum those products across all categories and flip the sign, since log2 of a proportion less than 1 is negative. Divide the result by log2 of the number of categories to get the normalized evenness score.
Examples
Five Unequal Categories
With counts of 40, 25, 20, 10, and 5 (100 total), the proportions are 0.40, 0.25, 0.20, 0.10, and 0.05. The calculator returns entropy of about 2.04 bits against a maximum of log2(5) = 2.32 bits, giving a normalized evenness of about 0.88.
Four Equal Categories
With counts of 25, 25, 25, and 25, every proportion is exactly 0.25. Entropy comes out to exactly 2.00 bits, which equals log2(4), so the normalized evenness score is a perfect 1.00.
Advantages
- Reduces a full distribution of category counts into one interpretable number, rather than requiring a visual scan of a table or chart.
- The normalized evenness score lets you compare diversity across datasets that have different numbers of categories, which raw entropy in bits cannot do on its own.
- Works directly from raw counts, so there is no need to manually convert to percentages first.
Common Mistakes
- Assuming entropy values are directly comparable across datasets with different category counts without checking the normalized score, since raw entropy is capped by log2(k) and that ceiling changes with k.
- Mixing up bits, the base-2 logarithm used here, with nats, a different but related convention using the natural logarithm in some other entropy tools.
- Treating a zero-count category as an error or undefined term instead of understanding it is simply excluded from the sum with no penalty.
Edge Cases to Watch For
- Needs counts for at least 2 categories; a single category returns an error since there is nothing to compare.
- Negative counts are rejected outright, since a proportion cannot be negative.
- Categories with a count of zero are skipped in the summation rather than causing an error, since log2(0) is undefined; this follows the standard convention that a zero-probability category contributes zero to entropy.
Common Use Cases
- Ecologists calculating a species diversity index from counts of individuals observed per species.
- Analysts assessing how concentrated or fragmented a market is based on counts of transactions or customers per firm.
- Data scientists and students exploring information theory concepts such as uncertainty and compressibility of categorical data.