Statistics Homework 2 — Datasets, Distributions & Cryptanalysis

Overview

This homework explores the foundational concepts of datasets and distributions in statistical analysis, followed by an application to cryptanalysis using frequency analysis. The assignment is structured in three parts:

1. Conceptual Foundations: Rigorous definitions of datasets, distributions, variables, and measurement scales
2. Database & Statistical Distributions: Construction of a relational dataset with computation of univariate and bivariate distributions
3. Cryptanalysis with Letter Frequencies: Caesar cipher encryption/decryption using statistical frequency analysis and automated language recognition

This assignment demonstrates how statistical methods underpin both descriptive data analysis and practical applications in information security, particularly in classical cryptanalysis and pattern recognition.

1. Datasets: Definition and Structure

1.1 What is a Dataset?

A dataset is a structured collection of observations organized in tabular form, where:

• Rows (observations, units, cases): Represent individual entities being studied (e.g., products, customers, network packets, malware samples)
• Columns (variables, attributes, features): Represent measurable or observable characteristics of each entity
• Cells (values, data points): Contain the specific measurement or observation for a given unit-variable pair

Formally, a dataset \(D\) with \(n\) observations and \(p\) variables can be represented as an \(n \times p\) matrix: \[ D = \begin{bmatrix} x_{1,1} & x_{1,2} & \cdots & x_{1,p} \\ x_{2,1} & x_{2,2} & \cdots & x_{2,p} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n,1} & x_{n,2} & \cdots & x_{n,p} \end{bmatrix} \] where \(x_{i,j}\) is the value of variable \(j\) for observation \(i\).

1.2 Components of a Dataset

Statistical Units: The indivisible entities being measured. Each row corresponds to exactly one unit. Units must be clearly defined to ensure data consistency (e.g., "a product in inventory" vs. "a product SKU" — these are different conceptual units).

Variables: Operational rules that map each unit to a value. Variables transform theoretical attributes (abstract properties) into measurable data. For example, the attribute "product size" becomes a variable when we define it as "length in centimeters" or "categorical size (Small/Medium/Large)".

Measurement Scales: The mathematical structure of a variable determines which statistical operations are valid:

Critical Principle: Statistical validity requires matching analytical methods to measurement scales. Computing the "average color" of products (nominal scale) is mathematically meaningless, whereas computing the average packet size (ratio scale) is both valid and interpretable.

1.3 Tidy Data Principles

Well-structured datasets follow the tidy data paradigm (Wickham, 2014):

1. Each variable forms a column: One column = one type of measurement
2. Each observation forms a row: One row = one complete unit
3. Each type of observational unit forms a table: Related data about different entities belong in separate tables linked by keys

This structure enables efficient querying, transformation, and analysis while maintaining data integrity.

1.4 Database Management Systems (DBMS)

A Database Management System provides structured storage and retrieval of datasets through:

• Schema definition: Specification of table structures, data types, constraints (primary keys, foreign keys, uniqueness, NOT NULL)
• Data integrity: Enforcement of referential integrity, domain constraints, and business rules
• Query language (SQL): Declarative specification of data retrieval and aggregation operations
• Transaction management: ACID properties (Atomicity, Consistency, Isolation, Durability) for reliable concurrent access

Relational DBMS (e.g., PostgreSQL, MySQL, SQLite) organize data into tables with relationships defined by foreign key constraints, enabling complex queries across multiple related datasets.

2. Distributions: From Data to Summary

2.1 Concept of a Distribution

While a dataset is a collection of individual observations, a distribution is a summary that describes the frequency or relative frequency with which different values occur. Distributions abstract away individual data points to reveal patterns in the data.

For a variable \(X\) with possible values \(\{x_1, x_2, \ldots, x_k\}\), the frequency distribution specifies:

• Absolute frequency \(n_i\): Count of observations with value \(x_i\)
• Relative frequency \(f_i = \frac{n_i}{n}\): Proportion of observations with value \(x_i\), where \(n = \sum_{i=1}^{k} n_i\)

The relative frequency distribution satisfies \(\sum_{i=1}^{k} f_i = 1\) and provides a discrete probability distribution for the observed data.

2.2 Univariate Distributions

A univariate distribution describes the behavior of a single variable. It answers: "How are the values of this variable spread across its possible range?"

For categorical variables (nominal/ordinal), univariate distributions are displayed as:

• Frequency tables: Listing each category with its count and percentage
• Bar charts: Visual representation of frequencies or relative frequencies

For continuous variables (interval/ratio), distributions use:

• Histograms: Binning continuous values into intervals and plotting frequencies
• Density plots: Smooth estimates of the probability density function
• Summary statistics: Mean, median, mode, standard deviation, quartiles

2.3 Bivariate Distributions

A bivariate distribution describes the joint behavior of two variables, revealing relationships and dependencies between them. It answers: "How do values of variable X associate with values of variable Y?"

For two categorical variables \(X\) and \(Y\), the bivariate distribution is represented as a contingency table (cross-tabulation):

• Rows correspond to categories of \(X\)
• Columns correspond to categories of \(Y\)
• Each cell \((i,j)\) contains the count or percentage of observations where \(X = x_i\) and \(Y = y_j\)

Contingency tables support several types of frequency analysis:

• Joint frequencies: \(n_{ij}\) = count of observations with \(X = x_i\) and \(Y = y_j\)
• Marginal frequencies: Row totals (\(\sum_j n_{ij}\)) and column totals (\(\sum_i n_{ij}\))
• Conditional frequencies: Row-wise percentages \(\frac{n_{ij}}{\sum_j n_{ij}}\) show distribution of \(Y\) given \(X = x_i\); column-wise percentages show distribution of \(X\) given \(Y = y_j\)

Conditional distributions reveal whether the distribution of one variable depends on the value of the other, suggesting association or independence.

2.4 Applications in Data Analysis

Distributions are fundamental to:

• Descriptive statistics: Summarizing and communicating data patterns
• Outlier detection: Identifying unusual observations that deviate from typical patterns
• Hypothesis testing: Comparing observed distributions to theoretical models (e.g., chi-squared goodness-of-fit tests)
• Data quality assessment: Detecting missing values, encoding errors, or implausible values
• Feature engineering: Transforming variables to improve model performance (e.g., normalization, binning)

3. Interactive Dataset: Inventory Management System

3.1 Dataset Design

We construct a simulated inventory dataset for a small retail business. Each observation represents a product item with the following variables:

• ID (integer, ratio scale): Unique identifier for each item (primary key)
• Item (text, nominal scale): Product type — Pen, Notebook, or Eraser
• Color (text, nominal scale): Product color — Red, Blue, or Green
• Size (text, ordinal scale): Product size category — Small, Medium, or Large (ordered)

This dataset demonstrates:

• A relational structure suitable for SQL databases
• Mixed measurement scales (nominal and ordinal categorical variables)
• Sufficient observations (\(n \geq 50\)) for meaningful frequency analysis

3.2 Generated Dataset

The dataset below is programmatically generated with systematic combinations and additional random samples to create realistic frequency patterns.

3.3 Univariate Distributions

Frequency distributions for each variable showing absolute counts, relative frequencies (percentages), and visual representations.

SQL Queries for Univariate Distributions

The following SQL queries compute frequency distributions for each variable in the inventory dataset.

SELECT 
    item,
    COUNT(*) AS Frequency,
    ROUND(COUNT(*) * 100.0 / (SELECT COUNT(*) FROM inventory), 2) AS RelativeFrequency
FROM inventory
GROUP BY item
ORDER BY Frequency DESC;

SELECT 
    color,
    COUNT(*) AS Frequency,
    ROUND(COUNT(*) * 100.0 / (SELECT COUNT(*) FROM inventory), 2) AS RelativeFrequency
FROM inventory
GROUP BY color
ORDER BY Frequency DESC;

SELECT 
    size,
    COUNT(*) AS Frequency,
    ROUND(COUNT(*) * 100.0 / (SELECT COUNT(*) FROM inventory), 2) AS RelativeFrequency
FROM inventory
GROUP BY size
ORDER BY Frequency DESC;

3.4 Bivariate Distribution: Item × Color

Contingency table showing the joint distribution of Item and Color with conditional (row-wise) percentages. This reveals whether certain colors are more common for specific product types.

SQL Query for Bivariate Distribution

This query computes the conditional distribution showing color distribution within each item type (row percentages).

SELECT 
    i.item,
    i.color,
    COUNT(*) AS Frequency,
    ROUND(COUNT(*) * 100.0 / t.ItemTotal, 2) AS ConditionalPercentage
FROM inventory i
JOIN (
    SELECT item, COUNT(*) AS ItemTotal
    FROM inventory
    GROUP BY item
) t ON i.item = t.item
GROUP BY i.item, i.color, t.ItemTotal
ORDER BY i.item, Frequency DESC;

4. Letter Frequency Analysis

4.1 Theoretical Foundation

Letter frequency analysis is a cornerstone technique in classical cryptanalysis. Natural language texts exhibit highly non-uniform distributions of letters, with certain letters (e.g., E, T, A in English; E, A, I in Italian) appearing much more frequently than others (e.g., Q, X, Z).

These characteristic frequency patterns remain invariant across sufficiently large texts in a given language, creating a statistical fingerprint that can be exploited to:

• Identify languages: Different languages have distinct frequency profiles
• Break substitution ciphers: Frequency patterns persist even after character substitution
• Assess text authenticity: Deviations from expected frequencies may indicate encryption, steganography, or non-textual data

4.2 Computing Letter Distribution

Given a text \(T\) of length \(n\) characters (ignoring spaces and punctuation), the letter frequency distribution is computed as: \[ f_i = \frac{c_i}{n} \times 100\% \] where \(c_i\) is the count of letter \(i\) (case-insensitive) and \(n = \sum_{i=A}^{Z} c_i\) is the total number of alphabetic characters.

This distribution \(\mathbf{f} = (f_A, f_B, \ldots, f_Z)\) forms a 26-dimensional probability vector representing the empirical frequency profile of the text.

4.3 Reference Language Profiles

Theoretical letter frequencies for major European languages (based on large corpora analysis):

Note: These are aggregate statistics. Individual texts may deviate due to topic-specific vocabulary (e.g., technical documentation vs. literary fiction).

4.4 Interactive Letter Frequency Analyzer

5. Caesar Cipher: Encryption and Decryption

5.1 Cipher Mechanism

The Caesar cipher is a monoalphabetic substitution cipher that shifts each letter by a fixed number of positions in the alphabet. Formally, for a shift parameter \(k \in \{0, 1, \ldots, 25\}\), the encryption function is: \[ E_k(x) = (x + k) \mod 26 \] where \(x \in \{0, 1, \ldots, 25\}\) represents the letter position (A=0, B=1, ..., Z=25).

The decryption function is the inverse operation: \[ D_k(y) = (y - k) \mod 26 \]

Key properties:

• Preservation of frequency distribution: The letter frequency pattern is preserved but permuted. If E is the most common letter in plaintext, the letter at position (E + k) mod 26 will be most common in ciphertext.
• Key space: Only 25 possible non-trivial keys (k=0 is identity transformation), making the cipher vulnerable to exhaustive key search (brute force)
• Historical significance: Named after Julius Caesar, who reportedly used shift k=3 for military communications

5.2 Statistical Attack via Frequency Analysis

The vulnerability of the Caesar cipher lies in its failure to mask statistical properties of the plaintext. The attack proceeds as:

1. Compute ciphertext frequency distribution \(\mathbf{f}_{cipher} = (f_A, f_B, \ldots, f_Z)\)
2. For each possible shift \(k \in \{0, 1, \ldots, 25\}\):
   • Shift the ciphertext frequencies back by \(k\) positions: \(\mathbf{f}_{shifted}(k) = (f_{(A-k) \mod 26}, f_{(B-k) \mod 26}, \ldots)\)
   • Compute similarity between \(\mathbf{f}_{shifted}(k)\) and reference language profile \(\mathbf{f}_{lang}\)
3. Select the shift \(k^*\) that maximizes similarity (minimizes distance)
4. Decrypt using \(k^*\)

5.3 Chi-Squared Distance Metric

To quantify the similarity between two frequency distributions, we use the chi-squared distance: \[ \chi^2(\mathbf{f}_{obs}, \mathbf{f}_{exp}) = \sum_{i=A}^{Z} \frac{(f_{obs,i} - f_{exp,i})^2}{f_{exp,i}} \] where:

• \(\mathbf{f}_{obs}\) is the observed frequency distribution (shifted ciphertext)
• \(\mathbf{f}_{exp}\) is the expected frequency distribution (language profile)

A smaller \(\chi^2\) value indicates better agreement between distributions. The shift that minimizes \(\chi^2\) is the most likely correct decryption key.

Alternative metrics: Other distance measures include Kullback-Leibler divergence, cosine similarity, and correlation coefficient. Chi-squared is preferred here for its statistical interpretation in goodness-of-fit testing.

5.4 Interactive Caesar Cipher Encryption

5.5 Interactive Automated Decryption

Paste encrypted text below. The system will attempt all 26 possible shifts, compute chi-squared distances against reference language profiles, and identify the most likely decryption.

6. Automated Language Recognition

6.1 Multi-Language Cryptanalysis

In realistic scenarios, the source language of encrypted text may be unknown. Automated language detection extends frequency-based cryptanalysis by:

1. Computing ciphertext frequency profile \(\mathbf{f}_{cipher}\)
2. For each candidate language \(L \in \{\text{English, Italian, Spanish, German, French}\}\):
   • For each shift \(k \in \{0, \ldots, 25\}\), compute \(\chi^2_L(k)\) between shifted frequencies and language profile
   • Record minimum distance \(\chi^2_{L,\text{min}} = \min_k \chi^2_L(k)\) and corresponding best shift \(k_L^*\)
3. Identify the language with smallest \(\chi^2_{L,\text{min}}\): \(L^* = \arg\min_L \chi^2_{L,\text{min}}\)
4. Decrypt using shift \(k_{L^*}^{*}\) for language \(L^*\)

6.2 Performance Considerations

The algorithm has computational complexity \(O(|L| \times 26 \times 26)\) where:

• \(|L|\) = number of candidate languages (typically 5-10)
• 26 possible shifts per language
• 26 letters in chi-squared computation per shift

For interactive applications, this is negligible (microseconds on modern hardware). However, for large-scale corpus analysis or real-time decryption, optimizations include:

• Language pre-filtering: Use simple heuristics (character sets, bigram frequencies) to narrow candidate languages
• Early termination: Stop testing shifts once a highly confident match (\(\chi^2 < \epsilon\)) is found
• Parallel evaluation: Test languages and shifts concurrently

6.3 Limitations and Extensions

This frequency-based approach has limitations:

• Short texts: Texts shorter than ~100 characters may not exhibit stable frequency patterns
• Mixed languages: Multilingual texts produce hybrid frequency profiles that don't match any single reference
• Specialized vocabulary: Technical jargon, proper nouns, or non-standard language may skew frequencies
• Polyalphabetic ciphers: More sophisticated ciphers (Vigenère, Enigma) require different cryptanalytic techniques

Extensions to improve robustness:

• Bigram/trigram analysis: Use two- or three-letter frequency patterns (e.g., "TH", "THE" in English)
• Word pattern matching: Exploit common words ("THE", "AND", "OF") with known letter patterns
• Bayesian inference: Combine frequency evidence with prior probabilities of languages and keys
• Machine learning: Train classifiers on large encrypted corpora for probabilistic language identification

6.4 Interactive Demo: Full Cryptanalysis Pipeline

This demo performs complete automated cryptanalysis: paste any Caesar-encrypted text, and the system will simultaneously identify the source language and decrypt the message without any prior knowledge of the key or language.

Conclusion

This homework demonstrated the fundamental relationship between data structure, statistical summaries, and practical applications:

• Part 1: Established rigorous definitions of datasets, distributions, and measurement scales as foundational concepts in statistical analysis
• Part 2: Implemented a relational database structure with computation of univariate and bivariate frequency distributions, illustrating how raw data is transformed into interpretable summaries
• Part 3: Applied frequency distribution analysis to classical cryptanalysis, demonstrating how statistical patterns in natural language enable automated decryption and language identification

The Caesar cipher cryptanalysis illustrates a broader principle: any system that preserves statistical structure is vulnerable to pattern-based attacks. Modern cryptography addresses this by designing ciphers (AES, ChaCha20) that produce output indistinguishable from random noise, eliminating frequency-based vulnerabilities.

Further Reading: For deeper exploration of frequency analysis in cryptanalysis, see Singh's "The Code Book" (1999), Kahn's "The Codebreakers" (1967), and modern treatments in Katz & Lindell's "Introduction to Modern Cryptography" (2020).