Enter probability distribution
Use decimal probabilities or percentages. Zero-probability outcomes remain visible but contribute no entropy.
Formula used
Shannon entropy: H(X) = −Σ p(xᵢ) logb[p(xᵢ)]
Here, p(xᵢ) is the normalized probability of outcome xᵢ. The base b sets the unit: base 2 gives bits, e gives nats, and 10 gives hartleys.
Self-information is I(xᵢ) = −logb[p(xᵢ)]. Each row contributes p(xᵢ) × I(xᵢ) to the total entropy. Zero-probability terms are treated as zero.
How to use this calculator
- Enter each possible state or event in the outcome column.
- Add its probability as a decimal or a percentage.
- Select strict validation or automatic normalization.
- Choose bits, nats, hartleys, or a valid custom base.
- Press Calculate entropy and inspect the result panel.
- Use the contribution table to identify uncertain outcomes.
- Download the values as CSV or a concise PDF report.
Example data table
Consider four measured detector states. Their probability total is one.
| Detector state | Probability | Self-information in bits | Entropy contribution in bits |
|---|---|---|---|
| State A | 0.40 | 1.322 | 0.529 |
| State B | 0.30 | 1.737 | 0.521 |
| State C | 0.20 | 2.322 | 0.464 |
| State D | 0.10 | 3.322 | 0.332 |
| Total entropy | 1.00 | — | 1.846 bits |
Understanding random variable entropy
Entropy measures spread
Entropy measures uncertainty in a random variable. A concentrated distribution has low entropy. One outcome dominates the observations. A balanced distribution has higher entropy. Several outcomes remain plausible. The calculation uses probability, not measured magnitude. A rare event contributes high self-information. Its weighted contribution can remain modest. Entropy therefore joins surprise with frequency. This makes it useful for measurements, communication systems, and statistical models.
Probability shape changes the result
Equal probabilities create the greatest entropy for a fixed number of states. Unequal probabilities reduce the result. When each state has probability 0.25, uncertainty is complete. When one state has probability 0.97, uncertainty becomes small. The calculator displays each row contribution clearly. Moderately likely events often contribute strongly. Very likely events carry little surprise. Very rare events are surprising, but their small weights limit influence.
Select units deliberately
Base two produces entropy in bits. The natural logarithm produces nats. Nats appear often in physics, thermodynamics, and mathematical modeling. Base ten gives hartleys. A custom base can support a specialized reporting convention. Changing the base changes the numerical value. It does not change the distribution itself. Keep the same base when comparing several experiments. Otherwise, comparisons can become misleading.
Validate the distribution
Probabilities must be nonnegative. Percentages should total 100 exactly. Strict validation catches unfinished distributions. Automatic normalization is useful for raw weights or rounded measurements. It rescales each valid weight by their total. Normalization preserves relative proportions. It cannot correct missing states or biased data. Add every meaningful outcome before calculating. Treat unknown states carefully. They can make true uncertainty larger than the reported value.
Read the advanced measures
Maximum entropy describes the equal-probability limit for states. Normalized entropy compares your result with that limit. Effective outcomes converts entropy into a state count. KL divergence measures departure from uniform balance. Redundancy is the remaining fraction below maximum entropy. Information standard deviation shows how much outcome surprise varies. They do not replace careful experimental design. Use them to compare repeated runs and detector modes.
Apply results in physics
Entropy calculations can describe uncertain spin states, photon outcomes, decay channels, sensor classifications, and discrete energy observations. They are not automatically thermodynamic entropy. Physical interpretation needs a clear model. Define the random variable first. Identify mutually exclusive outcomes. Estimate reliable probabilities. Then calculate entropy with a stated base. Report the unit alongside the value. This supports sound reproducible comparisons across studies.
Frequently asked questions
1. What does entropy of a random variable mean?
It measures average uncertainty across possible outcomes. Higher values indicate less predictable outcomes. The value depends on probabilities and the selected logarithm base.
2. Can I enter percentages?
Yes. Choose the percentage entry format, then enter values such as 25 instead of 0.25. Strict validation expects a total of 100 percent before conversion.
3. What happens when probabilities do not total one?
Strict validation returns an error. Automatic normalization rescales positive values so their decimal total becomes one. This is useful for relative weights and rounded measurements.
4. Does a zero probability affect entropy?
No. A zero-probability outcome contributes zero entropy. It can remain listed for documentation, but it is excluded from nonzero state counts and maximum entropy calculations.
5. Why are bits commonly used?
Bits use logarithm base two. They match binary decisions, digital communication, and many information-processing systems. One bit represents the uncertainty of two equally likely outcomes.
6. What are nats?
Nats use the natural logarithm. They are common in mathematical physics, probability theory, and statistical mechanics. One nat equals approximately 1.443 bits.
7. What is normalized entropy?
Normalized entropy divides observed entropy by maximum entropy for the nonzero outcomes. It ranges from zero to one. It makes distributions with different state counts easier to compare.
8. What does effective outcomes mean?
Effective outcomes is base raised to the entropy. It estimates how many equally likely states would create the same uncertainty. It is sometimes called perplexity.
9. Is this thermodynamic entropy?
It calculates Shannon entropy for a discrete random variable. Thermodynamic entropy requires physical state definitions, energy relationships, and appropriate constants. The concepts are related but not interchangeable.
10. Can I use a custom logarithm base?
Yes. Select Custom base and enter any positive value except one. Keep that base consistent whenever you compare entropy values from separate distributions.
11. What does the contribution chart show?
The chart compares normalized probability with each outcome's entropy contribution. It helps identify events that meaningfully shape the total uncertainty, rather than only the most surprising events.