> ## Documentation Index
> Fetch the complete documentation index at: https://docs.cipherinspector.com/llms.txt
> Use this file to discover all available pages before exploring further.

# Kolmogorov-Smirnov Test

> The Kolmogorov-Smirnov test measures how closely a ciphertext character distribution matches a reference distribution.

## How it works

The Kolmogorov-Smirnov (K-S) test is a statistical method that compares two probability distributions by measuring the maximum vertical
distance between their cumulative distribution functions (CDFs).

<Steps>
  <Step title="Character Frequency Counting">
    The widget first counts the frequency of each character (or n-gram) in your ciphertext. This creates an observed distribution of
    how often each character appears.
  </Step>

  <Step title="Reference Distribution">
    A reference distribution is selected based on your comparison mode:

    * **Uniform**: Equal probability for all observed characters
    * **English**: Standard English letter frequencies (e.g., E at 12.7%, T at 9.06%)
    * **Ciphertext**: The character distribution from another ciphertext you select
  </Step>

  <Step title="CDF Construction">
    Both the observed and reference distributions are converted into cumulative distribution functions. A CDF shows, for each
    character, the total probability of that character and all characters before it.
  </Step>

  <Step title="D-Statistic Calculation">
    The D-statistic is calculated as the maximum absolute difference between the two CDFs at any point. A smaller D-statistic
    indicates the distributions are more similar.
  </Step>

  <Step title="P-Value Calculation">
    The p-value represents the probability that the observed difference could occur by chance. Higher p-values suggest the
    distributions match well; lower p-values indicate significant differences.
  </Step>
</Steps>

## Comparison Modes

### Uniform Distribution

Compares your ciphertext against a uniform distribution where every character has equal probability. This is useful for detecting whether
encryption has produced evenly distributed output.

### English Frequencies

Compares your ciphertext against standard English letter frequencies. Only alphabetic characters (A-Z) are analyzed. This helps identify
if text resembles natural English.

### Another Ciphertext

Compares your ciphertext against the character distribution of a selected base ciphertext. Useful for determining if two ciphertexts were
encrypted using similar methods or share statistical properties.

## Display Modes

### Score View

Displays a table with:

* **D-statistic**: The maximum difference between CDFs (0 to 1 scale)
* **P-value**: Statistical significance (color-coded for quick interpretation)
* **Interpretation**: Human-readable assessment of the result
* **Sample size**: Number of characters or n-grams analyzed

### CDF Graph View

Displays an interactive chart showing:

* **Solid lines**: Observed CDF from your ciphertext
* **Dashed lines**: Expected CDF from the reference distribution
* **Dotted vertical line**: Location of maximum difference (D-statistic)

## N-gram Settings

### N-gram Size

Instead of analyzing single characters, you can group characters into n-grams:

* **1**: Single characters (default)
* **2**: Bigrams (pairs like "TH", "HE")
* **3**: Trigrams (triplets like "THE", "AND")

Larger n-grams capture patterns in character sequences but require longer texts for meaningful analysis.

### N-gram Mode

* **Sliding Window**: Overlapping n-grams (ABCD → AB, BC, CD)
* **Block**: Non-overlapping n-grams (ABCD → AB, CD)

## P-Value Interpretation

| P-value Range | Interpretation               |
| ------------- | ---------------------------- |
| > 0.10        | Distributions match well     |
| 0.05 - 0.10   | Slight deviation             |
| 0.01 - 0.05   | Significant deviation        |
| \< 0.01       | Very different distributions |

## Practical Applications

The Kolmogorov-Smirnov test can be used to:

* Determine if ciphertext has uniform byte distribution (suggesting strong encryption)
* Identify if plaintext resembles natural English
* Compare multiple ciphertexts to detect similar encryption methods
* Analyze whether a substitution cipher preserves frequency patterns

## Caveats

* English frequency comparison only analyzes alphabetic characters; non-alphabetic characters are filtered out
* For n-gram sizes greater than 1, English frequency comparison falls back to uniform distribution (no reference English n-gram
  frequencies available)
* Very short texts may produce unreliable p-values due to small sample sizes
* The test measures overall distribution similarity, not specific character mappings