Shannon Entropy Calculator

Enter Data

Example Data Table

Counts from four observed states in a measurement sequence.

With base 2, this distribution gives an entropy near 1.846 bits. A more uniform distribution produces a higher entropy value.

Formula Used

Shannon entropy measures uncertainty in a discrete distribution:

H = −∑ p_i log_b(p_i)

• p_i is the probability of state i.
• b is the logarithm base: 2 gives bits, e gives nats, 10 gives hartleys.
• If you enter counts, probabilities are computed as p_i = (c_i + ε) / ∑(c + ε).
• Maximum entropy for N states is H_max = log_b(N).

1) What entropy measures

Shannon entropy summarizes how unpredictable a discrete outcome is. When a system’s states occur with similar probabilities, uncertainty rises and the entropy increases. When one state dominates, outcomes become easier to predict and the entropy decreases. In experiments, this provides a compact descriptor of randomness in observed data.

2) Physical meaning in measured signals

In physics, entropy from a probability model can be applied to symbolic sequences, binned amplitudes, energy levels, or occupancy states. For time-series analysis, you often convert a continuous sensor stream into discrete bins, then compute the distribution of visits. Higher entropy can indicate broader exploration of state space or stronger noise.

3) Building probabilities from counts

Many datasets start as counts: how often each state appears in a run. This calculator converts counts to probabilities using p_i = (c_i + ε) / ∑(c + ε), where ε is optional smoothing. This approach supports histograms, categorical outcomes, and discretized trajectories while keeping the computation consistent across trials.

4) Choosing the logarithm base

The log base sets the reporting unit. Base 2 returns entropy in bits, common in digital sampling and coding. Base e returns nats, often convenient in analytical derivations. Base 10 gives hartleys, useful when comparing with decimal orders of magnitude. Changing base rescales values but preserves ranking across datasets.

5) Normalization, validity, and precision

If your entries are intended as probabilities, they should sum to one. When they come from imperfect normalization, enable the normalization option to avoid misleading entropy. Precision controls rounding in the displayed table and exports, which matters when comparing close conditions, repeated trials, or small changes during parameter sweeps.

6) Zeros and epsilon smoothing

Exact zeros are common in sparse distributions, especially with many possible states. You can ignore explicit zeros to simplify the table without changing entropy, because zero-probability states contribute nothing. If you want to avoid instability from extremely small values, add ε smoothing and renormalize to keep probabilities well behaved.

7) Maximum and normalized entropy

For N symbols, the maximum entropy is H_max = log_b(N), achieved by a uniform distribution. The normalized value H/H_max helps compare datasets with different numbers of states, such as changing bin counts in a histogram or varying alphabet sizes in symbolic dynamics.

8) Typical applications and reporting

Shannon entropy is used in turbulence proxies, complexity studies, experimental diagnostics, compressibility estimates, and quality checks for random-number sources. Export the contribution table to document which states drive uncertainty, and report the selected unit plus the number of symbols. For reproducibility, keep your binning rule and ε setting fixed.