Formula Used
Binary Shannon entropy uses one probability, p, and its complement, 1 − p.
H(p) = −p log₂(p) − (1 − p) log₂(1 − p)
When p is zero, the term p log₂(p) is treated as zero. This follows the standard limit. Entropy is highest at p = 0.5. It becomes zero when one outcome is certain.
The calculator also reports min entropy, collision entropy, surprisal, redundancy, perplexity, and information variance.
How to Use This Calculator
- Select the input type that matches your data.
- Enter probability, percentage, counts, or a bit string.
- Choose the main output unit.
- Add sample size if you want an approximate error check.
- Use smoothing alpha only for measured counts or bit strings.
- Press the calculate button.
- Download the result as CSV or PDF if needed.
Example Data Table
| p(1) |
p(0) |
Entropy Bits |
Meaning |
| 0.50 |
0.50 |
1.000000 |
Maximum uncertainty |
| 0.75 |
0.25 |
0.811278 |
Moderate bias |
| 0.90 |
0.10 |
0.468996 |
Strong bias |
| 0.99 |
0.01 |
0.080793 |
Near certainty |
Why Binary Entropy Matters
Binary Shannon entropy measures uncertainty in a two state system. The states may be yes and no. They may be one and zero. They may also represent spin up and spin down. A balanced system has the greatest uncertainty. A biased system has less uncertainty. This idea helps students compare signals, sensors, and noisy experiments.
Physics Link
Information is not only a computing idea. It appears in statistical mechanics, communication theory, and measurement. A binary measurement can carry at most one bit of entropy. That happens when both outcomes are equally likely. When one outcome dominates, the result becomes easier to predict. The entropy then falls toward zero. This calculator shows that change clearly.
Reading The Values
The main result is Shannon entropy in bits. It uses base two logarithms. You can also view nats and hartleys. The normalized value compares entropy with the one bit maximum. Redundancy shows the unused part of that maximum. Min entropy focuses on the most likely event. Collision entropy checks the chance that two independent samples match.
Practical Use
Choose the input type first. Use probability when you already know p. Use counts when you measured events. Use percent for survey style data. Use a bit string for raw binary output. The page converts each input into probabilities. It then calculates all results from the same model.
Good inputs are important. Probabilities must stay between zero and one. Counts cannot be negative. A bit string should contain only zeros and ones. If a probability is exactly zero, its entropy term is treated as zero. This follows the standard limit rule.
The table helps compare common cases. It shows how entropy rises near balance. It also shows why rare events carry high surprise. Use the export tools when saving lab notes. The CSV file is useful for spreadsheets. The PDF report is useful for sharing a quick result. Always connect the number with the physical source. Entropy describes uncertainty, not truth by itself.
For better judgment, test several simple inputs. Compare a fair source with a biased source. Note how small probability changes near balance can move the result. This daily habit makes entropy easier to understand in experiments and models.
FAQs
What is binary Shannon entropy?
It is a measure of uncertainty for two possible outcomes. The result is highest when both outcomes are equally likely. It is zero when one outcome is certain.
Why is the maximum value one bit?
A binary source has two possible states. With base two logarithms, a perfectly balanced binary source carries one bit of entropy per symbol.
Can I use counts instead of probabilities?
Yes. Enter the count of ones and zeros. The calculator converts those counts into probabilities before applying the Shannon entropy formula.
What does p(1) mean?
p(1) is the probability of the outcome labeled one. Its complement, p(0), is calculated as one minus p(1).
What are nats and hartleys?
Nats use natural logarithms. Hartleys use base ten logarithms. Bits use base two logarithms. They describe the same uncertainty in different units.
What is redundancy?
Redundancy is the unused part of the one bit maximum. A balanced source has low redundancy. A predictable source has high redundancy.
What is smoothing alpha?
Smoothing alpha adds a small prior amount to count-based inputs. It can reduce extreme estimates when measured data is very small.
Is this useful in physics?
Yes. Binary entropy is useful for measurements, spin states, noisy signals, detector outcomes, and simple statistical mechanics examples.