# Statistics Canonical URL: https://shipslides.com/d/mathematics-statistics Raw viewer URL: https://content.shipslides.com/d/mathematics-statistics/raw Category: Mathematics Slides: 23 Updated: 2026-05-17T20:51:30.525Z Tags: mathematics, statistics ## Summary The Science of Learning from Data Key sections include: Statistics; What Is Statistics ?; A Brief History; Probability : The Language of Uncertainty; Key Distributions; The Central Limit Theorem; Descriptive Measures; Sampling Methods; Hypothesis Testing; The p-Value Controversy. ## Slide Outline 1. Statistics 2. What Is Statistics ? 3. A Brief History 4. Probability : The Language of Uncertainty 5. Key Distributions 6. The Central Limit Theorem 7. Descriptive Measures 8. Sampling Methods 9. Hypothesis Testing 10. The p-Value Controversy 11. Confidence Intervals 12. Linear Regression 13. Beyond Simple Regression 14. ANOVA : Analysis of Variance 15. Bayesian Statistics 16. Experimental Design 17. Non-Parametric Methods 18. Correlation vs. Causation 19. The Replication Crisis 20. Statistics & Machine Learning 21. Statistics in Practice 22. Common Pitfalls 23. The Art of Uncertainty ## Slide Transcript ### Slide 1: Statistics - 𝓢 - The Science of Learning from Data - From probability theory to machine learning -- how we quantify uncertainty and extract meaning from noise - Probability - Inference - Regression - Bayesian - Hypothesis Testing - 1 / 23 ### Slide 2: What Is Statistics? - Statistics is the mathematical discipline concerned with collecting, organizing, analyzing, interpreting, and presenting data. It provides the tools to make decisions under uncertainty -- the universal condition of the real world. - Descriptive Statistics - Summarizes data that has already been collected. Tools include means, medians, standard deviations, histograms, and box plots. The goal: condense raw data into understandable patterns. - Inferential Statistics - Draws conclusions about a population based on a sample. Tools include confidence intervals, hypothesis tests, and regression models. The goal: generalize from the observed to the unobserved. - "All models are wrong, but some are useful." -- George E.P. Box, 1976 - 2 / 23 ### Slide 3: A Brief History - 1654 - Pascal and Fermat exchange letters on the "Problem of Points," founding probability theory. - 1713 - Jacob Bernoulli's Ars Conjectandi proves the Law of Large Numbers. - 1763 - Thomas Bayes's theorem published posthumously -- the foundation of Bayesian statistics. - 1809 - Gauss publishes the method of least squares and the normal distribution theory. - 1900-1930 - Karl Pearson, R.A. Fisher, Jerzy Neyman, and Egon Pearson create modern frequentist statistics: chi-squared test, ANOVA, maximum likelihood, p-values, confidence intervals. - 1950s-present - Computational revolution: bootstrap, MCMC, machine learning, and causal inference transform the field. - 3 / 23 ### Slide 4: Probability: The Language of Uncertainty - Probability is the mathematical foundation of statistics. It assigns a number between 0 and 1 to every possible event, where 0 means impossible and 1 means certain. - Three Interpretations - Classical - Probability = favorable outcomes / total outcomes. Works when outcomes are equally likely (dice, cards). Laplace's definition. - Frequentist - Probability = long-run relative frequency. Flip a coin 10,000 times; heads appears ~50%. Von Mises formalized this view. - Bayesian - Probability = degree of belief. Updated as evidence accumulates via Bayes' theorem. Subjective but rigorous. De Finetti championed this. - Kolmogorov's Axioms (1933) - P(A) ≥ 0  |  P(Ω) = 1  |  P(A ∪ B) = P(A) + P(B) if A ∩ B = ∅ - 4 / 23 ### Slide 5: Key Distributions - A probability distribution describes how the values of a random variable are spread. Choosing the right distribution is half the battle in applied statistics. - Discrete - Bernoulli: Single yes/no trial (coin flip) - Binomial: Number of successes in n trials - Poisson: Count of rare events (calls/hour, mutations/gene) - Geometric: Trials until first success - Continuous - Normal (Gaussian): Bell curve; heights, errors, IQ scores - Exponential: Time between Poisson events - Uniform: Equal probability across an interval - Student's t: Normal-like but heavier tails; small samples - Normal Distribution PDF - f(x) = (1 / σ√(2π)) · e-(x-μ)² / 2σ² - The normal distribution appears everywhere because of the Central Limit Theorem: the sum of many independent random variables tends toward normality regardless of their individual distributions. - 5 / 23 ### Slide 6: The Central Limit Theorem - Perhaps the most important theorem in all of statistics, the CLT explains why the normal distribution is so ubiquitous and why sampling works. - Statement - If X1, X2, ..., Xn are independent and identically distributed random variables with finite mean μ and variance σ², then as n → ∞, the distribution of the sample mean approaches a normal distribution N(μ, σ²/n). - Why It Matters - Sampling: You can estimate a population mean with a known margin of error - Polling: A random sample of ~1,000 can represent 330 million Americans with ~3% margin of error - Quality control: Manufacturing processes use CLT-based control charts to detect defects - Finance: Portfolio returns approximate normality when composed of many independent assets (though fat tails remain a trap) - Standardized Sample Mean - Z = (&Xbar; - μ) / (σ / √n) → N(0, 1) - 6 / 23 ### Slide 7: Descriptive Measures - Measures of Center - Mean: Arithmetic average. Sensitive to outliers. Sum/n. - Median: Middle value when sorted. Robust to outliers. Preferred for income, home prices. - Mode: Most frequent value. Useful for categorical data. - Trimmed mean: Drops top/bottom k% before averaging. Compromise between mean and median. - Measures of Spread - Range: Max minus min. Crude but intuitive. - Variance (s²): Average squared deviation from the mean. - Standard deviation (s): Square root of variance. Same units as data. - IQR: Q3 minus Q1. Used in box plots. Robust to outliers. - Shape - Skewness: Measures asymmetry. Positive = right tail, negative = left tail. Income is right-skewed; test scores often left-skewed. - Kurtosis: Measures tail heaviness. Normal = 3 (or 0 if excess kurtosis). Financial returns are leptokurtic ("fat-tailed"). - 7 / 23 ### Slide 8: Sampling Methods - The foundation of inferential statistics is the idea that a well-chosen sample can represent a much larger population. But how the sample is chosen matters enormously. - Probability Sampling - Simple random: Every member has equal chance (lottery, random number generator) - Stratified: Divide population into strata (age, region), sample from each proportionally - Cluster: Randomly select groups (schools, city blocks), sample everyone within - Systematic: Select every k-th member from a list - Non-Probability Sampling - Convenience: Whoever is available (college students, online volunteers) - Snowball: Participants recruit others (hard-to-reach populations) - Quota: Non-random selection meeting demographic targets - Purposive: Expert judgment selects "representative" cases - "The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a body of data." -- John Tukey, 1986 - 8 / 23 ### Slide 9: Hypothesis Testing - The most widely used (and misunderstood) framework in applied statistics. Developed by Fisher, Neyman, and E. Pearson in the 1920s-30s. - The Steps - State the null hypothesis (H0): no effect, no difference - State the alternative hypothesis (H1): an effect exists - Choose a significance level (α), typically 0.05 - Compute a test statistic (z, t, F, χ²) from your data - Find the p-value: the probability of seeing data this extreme if H0 is true - If p ≤ α, reject H0; otherwise, fail to reject - Type I Error (α) - Rejecting H0 when it is true. A "false positive." Controlled by choosing α. - Type II Error (β) - Failing to reject H0 when it is false. A "false negative." Power = 1 - β. - 9 / 23 ### Slide 10: The p-Value Controversy - The p-value has been called "the most used and abused statistical concept in science." In 2016, the American Statistical Association took the unprecedented step of issuing a formal statement on its interpretation. - What a p-Value IS - The probability of observing data at least as extreme as what was collected, assuming the null hypothesis is true. It measures the compatibility of the data with H0. - What a p-Value is NOT - The probability that H0 is true (that requires Bayes' theorem) - The probability that the result occurred by chance - A measure of effect size or practical significance - A binary "significant/not significant" verdict -- the 0.05 threshold is arbitrary - "The earth is round (p < .05)." -- Jacob Cohen, 1994 -- satirizing the ritual of null hypothesis testing - In 2019, over 800 scientists signed a letter in Nature calling for the retirement of "statistical significance" as a concept. The debate continues. - 10 / 23 ### Slide 11: Confidence Intervals - A confidence interval provides a range of plausible values for a population parameter, offering more information than a simple point estimate or p-value. - 95% CI for a Mean - x̄ ± z0.025 · (s / √n) = x̄ ± 1.96 · (s / √n) - Correct Interpretation - If we repeated the experiment many times, 95% of the resulting intervals would contain the true population parameter. Any single interval either contains the parameter or it does not -- we just don't know which. - Width Depends On - Sample size (n): Larger n → narrower interval - Variability (s): More variation → wider interval - Confidence level: Higher confidence (99% vs. 95%) → wider interval - 90%z = 1.645 - 95%z = 1.960 - 99%z = 2.576 - 11 / 23 ### Slide 12: Linear Regression - Regression models the relationship between a dependent variable (Y) and one or more independent variables (X). It is the workhorse of applied statistics. - Simple Linear Regression - Y = β0 + β1X + ε   where ε ~ N(0, σ²) - Key Concepts - β0 (intercept): Expected value of Y when X = 0 - β1 (slope): Change in Y for a one-unit increase in X - R²: Proportion of variance in Y explained by the model (0 to 1) - Residuals: Observed minus predicted values. Should be randomly scattered. - Least squares: Choose β0, β1 to minimize the sum of squared residuals - Assumptions (LINE) - Linearity, Independence of errors, Normality of residuals, Equal variance (homoscedasticity). Violating these can invalidate inference. - 12 / 23 ### Slide 13: Beyond Simple Regression - Multiple Regression - Y = β0 + β1X1 + β2X2 + ... + ε - Controls for confounders by including multiple predictors. Adjusted R² penalizes adding unnecessary variables. - Logistic Regression - For binary outcomes (yes/no, survived/died). Models the log-odds as a linear function of predictors. - log(p/(1-p)) = β0 + β1X - Polynomial & Spline - Captures nonlinear relationships by adding X², X³ terms or using piecewise smooth functions (splines). Flexible but risks overfitting. - Regularization - Ridge (L2): Shrinks coefficients. Lasso (L1): Shrinks and sets some to zero (variable selection). Elastic Net: Combines both. - 13 / 23 ### Slide 14: ANOVA: Analysis of Variance - ANOVA tests whether the means of three or more groups differ significantly. Developed by R.A. Fisher in the 1920s for agricultural experiments. - The Logic - ANOVA partitions total variability into two components: - Between-group variance: How much group means differ from the overall mean - Within-group variance: How much individual observations vary within each group - F-statistic - F = (Between-group variance) / (Within-group variance) = MSbetween / MSwithin - A large F means the groups differ more than expected by chance. If significant, post-hoc tests (Tukey's HSD, Bonferroni) identify which specific groups differ. - Variants - One-way ANOVA: One factor (e.g., drug A vs. B vs. C) - Two-way ANOVA: Two factors and their interaction (drug x dosage) - Repeated measures ANOVA: Same subjects measured multiple times - MANOVA: Multiple dependent variables simultaneously - 14 / 23 ### Slide 15: Bayesian Statistics - The Bayesian approach treats probability as a measure of belief, updated by evidence. Once controversial, it has become mainstream thanks to computational advances like Markov Chain Monte Carlo (MCMC). - Bayes' Theorem - P(H | D) = P(D | H) · P(H) / P(D) - Components - Prior P(H): What you believed before seeing data - Likelihood P(D|H): How probable the data is under H - Posterior P(H|D): Updated belief after seeing data - Evidence P(D): Normalizing constant - Advantages - Directly answers "What is the probability that H is true?" - Naturally incorporates prior knowledge - Handles small samples gracefully - Produces full posterior distributions, not just point estimates - Key tools: Stan, PyMC, BUGS/JAGS for computational Bayesian modeling. Used extensively in clinical trials, A/B testing, astrophysics, and natural language processing. - 15 / 23 ### Slide 16: Experimental Design - Good statistics begins before data collection. The design of an experiment determines what conclusions can be drawn. - Fisher's Principles (1935) - Randomization: Randomly assign subjects to treatment/control to eliminate confounders - Replication: Multiple independent observations to estimate variability - Blocking: Group similar subjects together to reduce noise - Modern Designs - Randomized Controlled Trial - The gold standard in medicine. Random assignment + blinding (single or double) + placebo control. Phase III drug trials typically enroll hundreds to thousands. - A/B Testing - The tech industry's version of the RCT. Users are randomly assigned to variant A or B. Netflix, Google, and Amazon run thousands of A/B tests per year. - Observational studies (where randomization is impossible) use techniques like matching, instrumental variables, and regression discontinuity to approximate causal inference. - 16 / 23 ### Slide 17: Non-Parametric Methods - When data violates the assumptions of parametric tests (normality, equal variance), non-parametric methods provide distribution-free alternatives. - Parametric → Non-Parametric - t-test → Mann-Whitney U (Wilcoxon rank-sum) - Paired t-test → Wilcoxon signed-rank - One-way ANOVA → Kruskal-Wallis - Pearson correlation → Spearman rank correlation - How They Work - Instead of using raw data values, non-parametric tests rank the observations and analyze the ranks. This makes them robust to outliers and skewed distributions. - Trade-off: they have less statistical power than parametric tests when parametric assumptions are met. - The Bootstrap (Efron, 1979) - Resample your data with replacement thousands of times to estimate the sampling distribution of any statistic -- mean, median, regression coefficients, anything. No distributional assumptions needed. One of the most influential statistical ideas of the 20th century. - 17 / 23 ### Slide 18: Correlation vs. Causation - The most important lesson in statistics: correlation does not imply causation. Two variables can be correlated because of a common cause, reverse causation, or pure coincidence. - Pearson Correlation Coefficient - r = Σ(xi - x̄)(yi - ȳ) / √[Σ(xi - x̄)² · Σ(yi - ȳ)²]   (-1 ≤ r ≤ 1) - Famous Spurious Correlations - Per capita cheese consumption correlates with death by bedsheet entanglement (r = 0.95) - Nicholas Cage films correlate with swimming pool drownings (r = 0.67) - Organic food sales correlate with autism diagnoses (both rising over time) - Causal Inference Tools - Randomized experiments: The gold standard for establishing causation - Instrumental variables: Find a "natural experiment" (Angrist & Imbens, Nobel 2021) - Difference-in-differences: Compare before/after changes in treatment vs. control groups - Directed acyclic graphs (DAGs): Judea Pearl's framework for causal reasoning - 18 / 23 ### Slide 19: The Replication Crisis - Beginning around 2011, science confronted a disturbing reality: many published findings could not be reproduced. - 36%Psychology studies replicated (Open Science Collaboration, 2015) - 11%Cancer biology studies fully replicated (eLife, 2021) - 70%Researchers who tried and failed to reproduce others' work (Nature survey, 2016) - Root Causes - P-hacking: Running many analyses and reporting only significant ones - HARKing: Hypothesizing After Results are Known - Publication bias: Journals prefer positive results; null findings go unpublished ("file drawer problem") - Small samples: Underpowered studies produce unreliable effect estimates - Reforms - Pre-registration of studies, registered reports, open data mandates, and the push for larger sample sizes and effect-size reporting are reshaping how science uses statistics. - 19 / 23 ### Slide 20: Statistics & Machine Learning - Machine learning grew out of statistics and computer science. The boundaries are blurry: many ML methods are rebranded statistical techniques. - Statistical Thinking - Emphasizes interpretability and inference - Models are hypothesis-driven - Focus on understanding why - Smaller, carefully designed datasets - Tools: R, SAS, Stata - ML Thinking - Emphasizes prediction accuracy - Models are data-driven - Focus on predicting what - Large-scale, messy datasets - Tools: Python, TensorFlow, PyTorch - Shared Foundations - Cross-validation, regularization, the bias-variance trade-off, maximum likelihood estimation, and information criteria (AIC, BIC) are core to both fields. Random forests are extensions of CART (Classification and Regression Trees), which is itself a statistical method. Neural networks are nonlinear regression models with many parameters. - 20 / 23 ### Slide 21: Statistics in Practice - Medicine - Clinical trials, survival analysis (Kaplan-Meier curves), meta-analysis. The FDA requires statistical evidence (usually p < 0.05 in two independent trials) before approving a drug. - Economics - Econometrics applies regression and time series to economic data. Instrumental variables, panel data models, and difference-in-differences are standard tools. Three Nobel Prizes in econometrics (2000, 2003, 2021). - Sports - "Moneyball" (2003) popularized sabermetrics. Expected goals (xG) in soccer, win probability models in baseball, and player tracking data have transformed every major sport. - Tech / Industry - A/B testing, recommendation engines, fraud detection, natural language processing, and quality control (Six Sigma). Google reportedly runs over 10,000 A/B tests per year on Search alone. - 21 / 23 ### Slide 22: Common Pitfalls - Even professionals fall into these traps. Statistical literacy means knowing what can go wrong. - Simpson's Paradox: A trend that appears in several groups reverses when the groups are combined. UC Berkeley gender bias case (1973): overall admission rates favored men, but department-by-department, women were admitted at higher rates. - Survivorship Bias: Analyzing only the successes. WWII planes: engineers reinforced where returning planes had bullet holes, until Abraham Wald pointed out they should reinforce where the holes were not -- those planes didn't return. - Base Rate Neglect: Ignoring the prevalence of a condition. A 99%-accurate test for a 1-in-10,000 disease will mostly produce false positives. - Ecological Fallacy: Inferring individual behavior from group-level data. States with high ice cream sales have high crime rates -- but individuals eating ice cream aren't committing crimes. - Overfitting: A model that fits the training data perfectly but predicts poorly on new data. Adding enough parameters can fit any dataset -- including noise. - Multiple Comparisons: Testing 20 hypotheses at α = 0.05 expects one false positive by chance. Bonferroni correction: divide α by the number of tests. - 22 / 23 ### Slide 23: The Art of Uncertainty - Statistics is not about eliminating uncertainty -- it is about quantifying it honestly. In a world drowning in data, statistical thinking is no longer optional. - 400+Years of theory - 2.5 QBBytes of data created daily - ∞Decisions to be made - "To call in the statistician after the experiment is done may be no more than asking him to perform a postmortem examination: he may be able to say what the experiment died of." - -- R.A. Fisher, 1938 - 23 / 23 ## Related Decks - [Topology](https://shipslides.com/d/mathematics-topology) - [Algebra](https://shipslides.com/d/mathematics-algebra) - [Applied Mathematics](https://shipslides.com/d/mathematics-applied-math) - [Calculus](https://shipslides.com/d/mathematics-calculus)