Statistics and Mathematics

Descriptive Statistics and Data Summarization

• What you Need to Know

  • Measures of Central Tendency

    • Mean, median, and mode calculations and interpretations
    • Weighted averages and trimmed means
    • When to use each measure based on data distribution
    • Resources:
      • Descriptive Statistics - Khan Academy - Central tendency and variability
      • Statistics with Python - SciPy statistical functions
      • Descriptive Statistics Guide - Penn State descriptive statistics

  • Measures of Variability and Distribution

    • Variance, standard deviation, and range
    • Quartiles, percentiles, and interquartile range
    • Skewness, kurtosis, and distribution shape
    • Resources:
      • Variability Measures - Khan Academy variability concepts
      • Distribution Analysis - SciPy statistical distributions
      • Statistical Summaries - Pandas descriptive statistics

  • Data Visualization for Exploration

    • Histograms, box plots, and distribution visualization
    • Scatter plots and correlation visualization
    • Summary statistics and data profiling
    • Resources:
      • Data Visualization Basics - Python visualization examples
      • Matplotlib Tutorial - Python plotting library
      • Seaborn Statistical Plots - Statistical data visualization

Probability Theory and Distributions

• What you Need to Know

  • Probability Fundamentals

    • Sample spaces, events, and probability rules
    • Conditional probability and Bayes' theorem
    • Independence and mutually exclusive events
    • Resources:
      • Probability - Khan Academy - Complete probability course
      • Think Bayes - Free Bayesian statistics book
      • Probability Course - MIT - MIT probability theory

  • Common Probability Distributions

    • Normal, binomial, and Poisson distributions
    • Exponential, gamma, and beta distributions
    • Distribution properties and parameter estimation
    • Resources:
      • Probability Distributions - Khan Academy distributions
      • SciPy Statistical Distributions - Python statistical functions
      • Distribution Gallery - Distribution relationships and properties

  • Central Limit Theorem and Sampling

    • Sampling distributions and standard error
    • Central Limit Theorem applications
    • Confidence intervals and margin of error
    • Resources:
      • Central Limit Theorem - Khan Academy CLT explanation
      • Sampling Methods - Penn State sampling techniques
      • Bootstrap Methods - ESL bootstrap chapter

Statistical Inference and Hypothesis Testing

• What you Need to Know

  • Hypothesis Testing Framework

    • Null and alternative hypotheses formulation
    • Type I and Type II errors understanding
    • P-values, significance levels, and statistical power
    • Resources:
      • Hypothesis Testing - Khan Academy hypothesis testing
      • Statistical Inference - Johns Hopkins statistical inference
      • P-value Interpretation - Penn State hypothesis testing

  • Common Statistical Tests

    • t-tests for mean comparisons
    • Chi-square tests for categorical data
    • ANOVA for multiple group comparisons
    • Resources:
      • Statistical Tests Guide - Python statistical testing
      • SciPy Statistical Tests - Statistical test implementations
      • Choosing Statistical Tests - Test selection guide

  • Effect Size and Practical Significance

    • Cohen's d and effect size interpretation
    • Confidence intervals for effect estimation
    • Statistical vs practical significance
    • Resources:
      • Effect Size - Effect size concepts and calculation
      • Confidence Intervals - Khan Academy confidence intervals
      • Statistical Power - Power analysis and sample size

Linear Algebra for Data Science

• What you Need to Know

  • Vectors and Vector Operations

    • Vector addition, scalar multiplication, and dot products
    • Vector norms and distance metrics
    • Vector projections and orthogonality
    • Resources:
      • Linear Algebra for Data Science - Fast.ai computational linear algebra
      • NumPy Linear Algebra - Linear algebra with NumPy
      • 3Blue1Brown Vectors - Visual vector explanation

  • Matrix Operations and Properties

    • Matrix multiplication and transpose operations
    • Matrix inverse and determinant calculations
    • Eigenvalues, eigenvectors, and matrix decomposition
    • Resources:
      • Matrix Operations - Khan Academy matrix operations
      • Eigenvalues and Eigenvectors - Visual eigenvalue explanation
      • Matrix Decomposition - Matrix decomposition for ML

  • Dimensionality Reduction Concepts

    • Principal Component Analysis (PCA) mathematical foundation
    • Singular Value Decomposition (SVD) applications
    • Linear transformations and feature space mapping
    • Resources:
      • PCA Tutorial - Principal Component Analysis explained
      • SVD Applications - MIT SVD tutorial
      • Dimensionality Reduction - Scikit-learn decomposition methods

Experimental Design and A/B Testing

• What you Need to Know

  • Experimental Design Principles

    • Randomized controlled trials and experimental controls
    • Sample size calculation and power analysis
    • Blocking, stratification, and confounding variables
    • Resources:
      • Experimental Design - Arizona State University experimental design
      • A/B Testing Course - Udacity A/B testing fundamentals
      • Design of Experiments - Penn State experimental design

  • A/B Testing Implementation

    • Test design and hypothesis formulation
    • Randomization and treatment assignment
    • Statistical analysis and result interpretation
    • Resources:
      • A/B Testing Guide - Optimizely A/B testing methodology
      • Experimental Analysis - Microsoft experimentation platform
      • Statistical Power Calculator - Sample size and power calculation

  • Advanced Experimental Techniques

    • Multi-armed bandit testing
    • Factorial designs and interaction effects
    • Quasi-experimental methods and observational studies
    • Resources:
      • Multi-Armed Bandits - Bandit algorithm implementations
      • Causal Inference - Causal inference methods
      • Quasi-Experiments - University of Pennsylvania causality course

Bayesian Statistics and Advanced Methods

• What you Need to Know

  • Bayesian Inference Fundamentals

    • Prior and posterior distributions
    • Bayes' theorem applications in data analysis
    • Credible intervals vs confidence intervals
    • Resources:
      • Bayesian Statistics - Duke University Bayesian methods
      • Think Bayes - Bayesian statistics with Python
      • PyMC Documentation - Probabilistic programming in Python

  • Markov Chain Monte Carlo (MCMC)

    • MCMC sampling methods and convergence
    • Gibbs sampling and Metropolis-Hastings algorithms
    • Bayesian model fitting and diagnostics
    • Resources:
      • MCMC Tutorial - MCMC sampling explanation
      • PyMC Tutorial - Bayesian modeling with PyMC
      • Bayesian Analysis with Python - Bayesian analysis examples

Time Series Analysis

• What you Need to Know

  • Time Series Components and Decomposition

    • Trend, seasonality, and cyclical patterns
    • Time series decomposition methods
    • Stationarity testing and transformation
    • Resources:
      • Time Series Analysis - Forecasting: Principles and Practice (free online)
      • Time Series with Python - Statsmodels time series analysis
      • Pandas Time Series - Time series functionality

  • Forecasting Methods

    • ARIMA models and Box-Jenkins methodology
    • Exponential smoothing techniques
    • Seasonal forecasting and trend analysis
    • Resources:
      • ARIMA Modeling - ARIMA implementation in Python
      • Forecasting Tutorial - Time series forecasting methods
      • Prophet Forecasting - Facebook's forecasting tool

Ready to Analyze Data? Continue to Module 2: Data Analysis to master data manipulation, cleaning, and exploratory analysis techniques.