Statistical Analysis

Statistical Analysis

Statistical Methods and Tests

Descriptive Statistics

• Measures of Central Tendency: Mean, median, mode
• Measures of Dispersion: Variance, standard deviation, range, interquartile range
• Distribution Shape: Skewness, kurtosis
• Correlation: Pearson, Spearman, Kendall correlation coefficients
• Covariance: Measure of joint variability

Probability Distributions

• Normal Distribution: Bell curve, symmetric, defined by mean and standard deviation
• Binomial Distribution: Number of successes in n independent trials
• Poisson Distribution: Number of events in fixed interval
• Exponential Distribution: Time between events in Poisson process
• Chi-Square Distribution: Sum of squared normal variables

Inferential Statistics

• Confidence Intervals: Range of values likely to contain population parameter
• Hypothesis Testing: Formal procedure for testing claims about populations
• p-values: Probability of observing results as extreme as current, assuming null hypothesis
• Statistical Power: Probability of correctly rejecting false null hypothesis
• Effect Size: Magnitude of difference or relationship

Hypothesis Testing

Hypothesis Structure

• Null Hypothesis (H0): Default assumption, no effect or difference
• Alternative Hypothesis (H1): Claim to be tested, effect or difference exists
• Type I Error: Rejecting true null hypothesis (false positive)
• Type II Error: Failing to reject false null hypothesis (false negative)
• Significance Level (α): Threshold for rejecting null hypothesis (typically 0.05)

Common Statistical Tests

• t-test: Compare means between two groups
  • One-sample t-test: Compare sample mean to known value
  • Independent t-test: Compare means of two independent groups
  • Paired t-test: Compare means of paired samples
• ANOVA: Compare means across multiple groups
  • One-way ANOVA: Single factor
  • Two-way ANOVA: Two factors with interaction
• Chi-Square Test: Test independence between categorical variables
• Mann-Whitney U Test: Non-parametric alternative to t-test
• Kruskal-Wallis Test: Non-parametric alternative to ANOVA

Multiple Testing Correction

• Bonferroni Correction: Divide α by number of tests
• False Discovery Rate (FDR): Control proportion of false positives
• Benjamini-Hochberg: Adaptive FDR control

A/B Testing Frameworks

Experimental Design

• Control Group: Receives current version or no treatment
• Treatment Group: Receives new version or treatment
• Random Assignment: Randomly assign subjects to groups
• Sample Size Calculation: Determine required sample size for desired power
• Stratification: Balance groups on important covariates

Metrics Selection

• Primary Metric: Main measure of success
• Secondary Metrics: Additional measures of interest
• Guardrail Metrics: Ensure no negative impact on important KPIs
• Binary Metrics: Conversion, click-through rate
• Continuous Metrics: Revenue, time on page

Statistical Significance

• Two-tailed Test: Test for difference in either direction
• One-tailed Test: Test for difference in specific direction
• Confidence Intervals: Provide range of plausible values
• Minimum Detectable Effect (MDE): Smallest effect detectable with given power

Common Pitfalls

• Peeking: Checking results before experiment ends
• Simpson's Paradox: Trend appears in groups but disappears when combined
• Novelty Effect: Temporary effect due to newness
• Selection Bias: Non-random assignment to groups

Time Series Analysis

Time Series Components

• Trend: Long-term increase or decrease
• Seasonality: Regular, predictable patterns
• Cyclical: Irregular, long-term cycles
• Irregular/Noise: Random fluctuations

Stationarity

• Definition: Statistical properties constant over time
• Tests: Augmented Dickey-Fuller (ADF), KPSS test
• Transformations: Differencing, log transformation
• Importance: Required for many time series models

Forecasting Methods

• Naive Forecast: Use last observed value
• Moving Average: Average of last n values
• Exponential Smoothing: Weighted average with decreasing weights
• ARIMA: AutoRegressive Integrated Moving Average
• Prophet: Facebook's forecasting tool for business time series
• Neural Networks: LSTM, GRU for complex patterns

Seasonal Decomposition

• Additive Model: Y = Trend + Seasonal + Residual
• Multiplicative Model: Y = Trend × Seasonal × Residual
• STL Decomposition: Seasonal-Trend decomposition using LOESS

Experimental Design

Design Principles

• Randomization: Random assignment to treatment groups
• Replication: Repeat experiment multiple times
• Blocking: Group similar experimental units together
• Factorial Design: Test multiple factors simultaneously
• Control Groups: Baseline for comparison

Sample Size Determination

• Power Analysis: Calculate required sample size
• Effect Size: Expected magnitude of effect
• Significance Level: Acceptable Type I error rate
• Power: Desired probability of detecting effect (typically 0.8)

Experimental Validity

• Internal Validity: Causal relationship between treatment and outcome
• External Validity: Generalizability to other populations/settings
• Construct Validity: Measurement accurately reflects concept
• Statistical Conclusion Validity: Appropriate statistical methods

Common Designs

• Completely Randomized Design: Random assignment to groups
• Randomized Block Design: Block on nuisance variables
• Factorial Design: Multiple factors with all combinations
• Crossover Design: Subjects receive multiple treatments
• Split-Plot Design: Hierarchical randomization