Calculator Inputs
Enter symbols with matching frequencies or probabilities. The tool sorts items by descending probability, builds a Shannon-Fano codebook, and evaluates coding quality.
Example Data Table
This sample shows weighted symbols commonly used to demonstrate code construction and average-length calculations.
| Symbol | Frequency | Probability | Typical Outcome |
|---|---|---|---|
| A | 30 | 0.3000 | Shortest code due to highest weight |
| B | 25 | 0.2500 | Short code with low contribution penalty |
| C | 15 | 0.1500 | Mid-length code |
| D | 12 | 0.1200 | Mid-length code |
| E | 10 | 0.1000 | Longer code |
| F | 8 | 0.0800 | Longer code due to lower weight |
Formula Used
Shannon-Fano coding sorts symbols in descending probability order, then repeatedly splits the list into two groups with nearly equal probability mass. The left group receives a 0, and the right group receives a 1. Repeating that partition builds each prefix code.
- Probability:
p(x) = w(x) / Σwwhen normalization is enabled. - Self-information:
I(x) = -log₂(p(x)) - Entropy:
H = -Σ p(x) log₂(p(x)) - Average code length:
L = Σ p(x) l(x) - Efficiency:
η = H / L × 100% - Redundancy:
1 - H/Lor100% - efficiency - Kraft sum:
Σ 2^-l(x), useful for prefix-code checking.
The message section multiplies each observed symbol by its assigned code length, then compares the total bits against a fixed-length code using ceil(log₂ n) bits per symbol.
How to Use This Calculator
- Choose whether the numeric list represents frequencies or probabilities.
- Enter one unique symbol for each numeric value.
- Enable normalization when values do not already sum to one.
- Optionally enter a message to test real encoded length.
- Click Calculate Coding Metrics to generate the codebook.
- Review entropy, average length, efficiency, redundancy, and the code table.
- Use the export buttons to save CSV or PDF copies.
Frequently Asked Questions
1. What does Shannon-Fano coding do?
It builds variable-length prefix codes from symbol probabilities. Higher-probability symbols receive shorter codes, which reduces average bit usage compared with equal-length coding when the distribution is uneven.
2. How is Shannon-Fano different from Huffman coding?
Both use variable-length prefix codes, but Huffman coding is guaranteed optimal for minimum average length. Shannon-Fano uses recursive probability partitioning, which is simpler conceptually but can be slightly less efficient.
3. Should I enter frequencies or probabilities?
Use frequencies when you have raw counts from data. Use probabilities when your values already represent normalized likelihoods. The normalization option converts any positive weighted list into valid probabilities automatically.
4. Why must every symbol be unique?
A codebook maps one code to one source symbol. Duplicate labels create ambiguity during encoding and decoding, so unique symbols are necessary for correct results.
5. What does entropy tell me here?
Entropy measures the theoretical lower bound on average bits needed per symbol for the given distribution. It helps you judge how close the generated Shannon-Fano code comes to ideal compression.
6. What is the Kraft sum used for?
The Kraft sum checks whether code lengths satisfy a prefix-code feasibility condition. A valid binary prefix code has a Kraft sum less than or equal to one, with equality in many complete trees.
7. Why does my message show an unknown symbol warning?
That warning appears when the message includes a symbol not listed in the input set. Add the missing symbol to the table or remove it from the message before recalculating.
8. When is Shannon-Fano coding useful?
It is useful for learning source coding, testing probability-driven compression ideas, comparing code efficiency, and exploring how uneven symbol distributions affect average transmitted or stored bit length.