Entropy Calculator - Shannon Entropy of a Dataset

What is an entropy calculator?

An entropy calculator finds the Shannon entropy of a set of numbers. Shannon entropy is a way to measure how unpredictable a dataset is. A dataset where every value is the same has zero entropy, because there is nothing uncertain about it. A dataset where every value is equally likely to appear has the highest possible entropy for its size.

The idea comes from information theory, a field started by the American mathematician Claude Shannon in 1948. Shannon wanted to measure how much information a message carries. He defined entropy as the average amount of "surprise" in a sequence of symbols. The same formula now shows up in data science, cryptography, biology, and machine learning, anywhere someone needs to measure randomness in a set of outcomes.

Shannon entropy formula

For a dataset with unique values x₁ through xₙ, each appearing with probability p(xᵢ), the Shannon entropy H is:

$H(X) = -\sum_{i=1}^{n} p(x_i) \log_2 p(x_i)$

In words: for each unique value, multiply its probability by the base-2 logarithm of that probability, add up all these products, then flip the sign. The result is always zero or positive.

This calculator always uses base-2 logarithms, so the result is measured in bits. Other bases exist for other purposes: natural logarithm gives units called nats, and base-10 gives units called hartleys. Bits are the standard unit in computing and information theory, which is why this calculator sticks to base 2.

Why the result can't be negative

Every probability p(xᵢ) is between 0 and 1, so its logarithm is zero or negative. Multiplying a probability by a negative or zero logarithm gives a negative or zero number. Summing those and flipping the sign always produces a result of zero or more.

Maximum possible entropy

For a dataset with n unique values, entropy is highest when every value appears equally often. That maximum equals log₂(n) bits. A dataset with 4 equally common unique values can reach at most 2 bits of entropy, because log₂(4) = 2. Any uneven distribution of the same 4 values gives a lower entropy.

How to calculate entropy: step by step

1. List the unique values in the dataset and count how many times each one appears.
2. Divide each count by the total number of values to get the probability of each unique value.
3. Take the base-2 logarithm of each probability, then multiply it by that same probability.
4. Add up all these products, then multiply the total by −1.

This calculator does the same four steps automatically. Type numbers into the input box, separated by spaces or commas, choose the matching format, and the entropy, probability table, and a bar chart appear right away. A table below the result shows the value, count, probability, and p(x) × log₂(p(x)) for each unique number, so the working is visible, not just the final answer.

Input rules

• Only numeric values are accepted: whole numbers, decimals, and negative numbers all work.
• Values are separated by spaces (example: 1 2 3 4) or by commas (example: 1,2,3,4), depending on the selected format.
• A dataset can hold up to 100,000 values. Entering more than that produces an error message asking for a smaller dataset.
• Text, symbols, or empty entries between separators are rejected with an error, rather than silently ignored.

Worked example

Take the dataset 1 2 3 1 2 1, which has six numbers.

First, count each unique value:

Next, apply the formula to each row and add the results:

$H = -(0.5 \log_2 0.5 + 0.3333 \log_2 0.3333 + 0.1667 \log_2 0.1667)$ $H = -(0.5 \times -1 + 0.3333 \times -1.585 + 0.1667 \times -2.585)$ $H \approx 1.4591 \text{ bits}$

The dataset has 3 unique values, so the maximum possible entropy is log₂(3) ≈ 1.585 bits. The actual result, 1.4591 bits, is lower than that maximum because the value 1 appears more often than the others, making the dataset slightly less random than a perfectly even split.

A dataset with no uncertainty

The dataset 5 5 5 5 5 has only one unique value, so its probability is 1. Since log₂(1) = 0, every term in the sum is zero, and the entropy is exactly 0 bits. There is nothing uncertain about a dataset where every value is identical.

Reading the result

• Entropy near 0 means the data is repetitive and predictable. One or a few values dominate.
• Entropy near log₂(n), where n is the count of unique values, means the data is close to evenly spread across all its unique values.
• Entropy of exactly 0 means every value in the dataset is the same.

Entropy on its own does not say whether a dataset is "good" or "bad." A password generator wants high entropy, because that makes the password hard to guess. A sensor that should read a constant temperature wants low entropy, because that means the reading is stable.

Where Shannon entropy is used

• Machine learning: decision tree algorithms use entropy to decide which feature best splits a dataset into predictable groups.
• Data compression: entropy sets the theoretical limit on how small a file can be compressed without losing information.
• Cryptography: entropy measures how unpredictable a password or a cryptographic key is.
• Genetics: entropy can highlight unusual or highly variable regions in a DNA sequence.
• Text analysis: treating letters or words as the "values" lets entropy measure how predictable a piece of text is.

Frequently asked questions

What is entropy in information theory? It is a number that measures how uncertain or unpredictable a dataset is. It is calculated from the probabilities of each unique value in the data, not from the values themselves.

How do you calculate Shannon entropy by hand? Count how often each unique value occurs, divide each count by the total to get probabilities, multiply each probability by its base-2 logarithm, add the results together, and multiply by −1.

Can entropy be negative? No. The lowest possible value is 0 bits, which happens when every value in the dataset is identical.

What is the maximum entropy for a dataset? The maximum is log₂(n) bits, where n is the number of unique values, and it only occurs when every unique value appears equally often.

Is there a limit on dataset size? Yes. This calculator accepts up to 100,000 values in a single dataset. Larger inputs return an error.

How is entropy different from variance? Variance measures how spread out numeric values are around their average. Entropy measures how unpredictable the pattern of outcomes is, based only on probabilities, regardless of the actual size of the numbers.

References

1. Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423.
2. Cover, T. M., & Thomas, J. A. (2006). Elements of Information Theory (2nd ed.). Wiley-Interscience.
3. MacKay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press.