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All 215 Statistics Fundamentals interview questions

Probability, hypothesis testing, confidence intervals, and the stats DS interviewers ask about.

Every question below opens on its own page, with a full answer, the formulas written out and a multiple-choice check. Free to read, no account needed.

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Probability foundations

  1. State the Central Limit Theorem in one sentence.easy
  2. Bayesian vs frequentist — what's the core difference?medium
  3. State the Law of Large Numbers.easy
  4. Independence vs uncorrelatedness — what's the difference?medium
  5. State Bayes' theorem and one intuitive use.easy
  6. State Kolmogorov's three probability axioms.easy
  7. Define conditional probability and prove Bayes' rule from it.easy
  8. State the law of total probability.medium
  9. Bernoulli distribution: parameters, PMF, mean, variance.easy
  10. Binomial distribution and when to use it.easy
  11. Poisson distribution and its typical use cases.medium
  12. Geometric distribution: setup and mean.medium
  13. Uniform distribution: continuous vs discrete.easy
  14. Exponential distribution: setup, memorylessness, use cases.medium
  15. Normal distribution: PDF and key properties.easy
  16. Multivariate normal: parameters and key properties.medium
  17. What is a covariance matrix and its key properties?medium
  18. What is linearity of expectation?easy
  19. Variance of a sum: Var(X + Y) = ?medium
  20. Write covariance and correlation formulas.easy
  21. What is E[X | Y] and its Law of Total Expectation?medium
  22. Law of Total Variance — Var(X) = ?hard
  23. Why does the Cauchy distribution have no mean?hard
  24. State Jensen's inequality and give an ML example.hard
  25. What is a moment generating function and why care?hard
  26. State Chebyshev's inequality.medium
  27. State Markov's inequality.medium
  28. What is Hoeffding's inequality?hard
  29. How does Monte Carlo estimation work and its convergence rate?medium
  30. What statistical properties make maximum likelihood the default estimator?medium
  31. What is Fisher information?hard
  32. Bias, variance, consistency of estimators — define.medium
  33. MLE for a Bernoulli(p) — derive.medium
  34. MLE for Normal(μ, σ²) — result.medium
  35. When would you use Hoeffding's inequality for a CI?hard
  36. Cramér-Rao lower bound — state it.hard
  37. What is the influence function?hard
  38. Give the OLS closed-form and its variance.medium
  39. State Bayes' theorem and what each term means.easy
  40. Frequentist vs Bayesian — key philosophical difference.medium
  41. Potential outcomes framework — Rubin causal model.hard

Common distributions

  1. Log-normal distribution and its uses.medium
  2. What is a power-law distribution and how do you spot one?medium
  3. Beta distribution: setup and use case.medium
  4. Gamma distribution and when it appears.medium
  5. Dirichlet distribution — where does it show up in ML?hard

Expectation & variance

  1. Why do we use standard deviation instead of variance in interpretation?easy
  2. Skewness and kurtosis — what do they measure?medium
  3. Why divide by n-1 in the sample variance?easy

Descriptive statistics & EDA

  1. Mean vs median vs mode — when do you prefer each?easy
  2. What are quantiles and quartiles?easy
  3. How do you read a boxplot?easy
  4. How do you choose bin width for a histogram?medium
  5. What is kernel density estimation?medium
  6. How do you define an outlier in practice?medium
  7. What is MAD and why is it robust?medium
  8. Describe a good EDA workflow for a new dataset.medium
  9. MCAR vs MAR vs MNAR — what's the difference?hard
  10. What imputation methods should you consider?medium
  11. When and why do you log-transform a variable?easy
  12. What is the Box-Cox transformation?medium
  13. Pearson vs Spearman vs Kendall correlation — when do you use each?easy
  14. What is an ECDF and why is it useful?medium
  15. How do you read a Q-Q plot?medium
  16. Standardization vs normalization — what's the difference?easy
  17. What multivariate EDA plots are most useful?medium
  18. How does EDA help catch target leakage?medium
  19. Why should you always do group-wise EDA?medium
  20. What time-series-specific EDA should you do?medium
  21. What do you look for in residual plots?medium
  22. When should you log-transform the response?medium
  23. Overall conversion went down but improved in every country. How is that possible?hard
  24. How do you spot selection bias in a dataset someone hands you?hard

Sampling & central limit theorem

  1. What is sampling bias and how do you mitigate it?easy
  2. Your revenue-per-user metric is extremely right-skewed. Can you still use a t-test?hard

Hypothesis testing

  1. What is a p-value, precisely?easy
  2. Type I vs Type II error — what's the difference?easy
  3. Why is multiple testing a problem and how do you correct for it?medium
  4. How do you formulate null and alternative hypotheses?easy
  5. One-sample t-test: setup and assumptions.easy
  6. Two-sample t-test: pooled vs Welch — when to use each?medium
  7. When do you use a paired t-test?easy
  8. z-test vs t-test — when to pick each?easy
  9. Chi-square test of independence — what does it do?medium
  10. Chi-square goodness-of-fit test — setup.medium
  11. What does one-way ANOVA test?medium
  12. Two-way ANOVA and what interactions mean.hard
  13. Mann-Whitney U (Wilcoxon rank-sum) — when and why?medium
  14. Wilcoxon signed-rank test — setup.medium
  15. Kruskal-Wallis — non-parametric ANOVA analog.medium
  16. Friedman test and its use case.hard
  17. Kolmogorov-Smirnov test — one- and two-sample.medium
  18. Shapiro-Wilk normality test — when useful?medium
  19. How do you test equality of variances?medium
  20. Why report effect size alongside p-values?easy
  21. What is power analysis and how do you use it?medium
  22. What is MDE (minimum detectable effect)?medium
  23. Bonferroni correction — how does it work?medium
  24. How does the Benjamini-Hochberg procedure work?hard
  25. What is a permutation test?medium
  26. What does McNemar's test do?hard
  27. When to use Fisher's exact test?medium
  28. What is p-hacking and how do you prevent it?medium
  29. FWER vs FDR — which do you control when?hard
  30. How is a CI related to a hypothesis test?medium
  31. Why is peeking at running A/B tests dangerous?hard
  32. What is alpha-spending in sequential testing?hard
  33. A product manager asks what a p-value of 0.03 means. What do you say?easy

Non-parametric tests

  1. How many bootstrap resamples do you need, and which statistics does it handle?medium

Power, effect size & multiple testing

  1. What techniques reduce variance in A/B tests?hard
  2. How does CUPED reduce variance in A/B tests?hard
  3. Why is checking an A/B test daily and stopping when it turns significant wrong?medium
  4. What do you need to know to compute a sample size for an experiment?medium
  5. Your experiment tracks 20 metrics and one is significant. What now?medium

Estimation: MLE, MoM, MAP

  1. What does a 95% confidence interval mean?medium
  2. What is the bootstrap and when do you use it?medium
  3. What is Maximum Likelihood Estimation?medium
  4. What is the Method of Moments (MoM)?medium
  5. MAP vs MLE — key difference.medium
  6. Explain the EM algorithm.hard
  7. Standard error vs standard deviation — the difference.easy
  8. What is the delta method?hard
  9. What is the jackknife?hard
  10. What is a conjugate prior?hard
  11. Beta-Bernoulli conjugate update — derive.medium
  12. What is an uninformative prior?hard
  13. What is Bayesian shrinkage?hard
  14. What is empirical Bayes?hard
  15. How does Thompson sampling work?medium
  16. Why do conjugate priors matter in production?medium
  17. What does a 95% confidence interval actually assert?medium
  18. When is a Bayesian analysis genuinely worth the extra effort?medium

Confidence intervals & bootstrap

  1. 95% CI for a mean — formula and interpretation.easy
  2. What CI methods exist for a proportion?medium
  3. Bootstrap CI methods — percentile vs BCa vs studentized.hard
  4. Why do you need block bootstrap for time series?hard
  5. Prediction interval vs confidence interval — the difference.medium
  6. Credible interval vs confidence interval.medium
  7. Why do ratio metrics need the delta method?hard
  8. When does the bootstrap give you the wrong answer?hard

Regression assumptions

  1. What are the Gauss-Markov assumptions for OLS?medium
  2. Heteroscedasticity — what breaks and how do you fix it?medium
  3. How do you detect and handle multicollinearity?medium
  4. R² vs adjusted R² — when do you use each?easy
  5. Cook's distance, leverage, DFBETAS — define.hard
  6. Standardized vs studentized residuals — the difference.hard
  7. Logistic regression: why MLE and not OLS?medium
  8. Odds ratio — interpret and derive.medium
  9. GLM: link function and exponential family — connection.hard
  10. Poisson regression: setup and pitfalls.hard
  11. Negative binomial regression — why use it?hard
  12. What is quasi-likelihood?hard
  13. What is a Generalized Additive Model (GAM)?hard
  14. Splines: knots, degrees of freedom, regularization.hard
  15. How do you include and interpret interactions in regression?medium
  16. What is a mixed-effects model?hard
  17. Random intercepts vs random slopes — the difference.hard
  18. Cluster-robust standard errors — when to use?hard
  19. Mixed-effects vs cluster-robust SEs — which do you pick?hard
  20. In practice, how do you handle correlated features in regression?medium
  21. Durbin-Watson test — what does it test?hard
  22. Two-stage least squares (2SLS) — the recipe.hard
  23. Your model reports R^2 = 0.92. What can go wrong with celebrating that?medium
  24. Two predictors correlate at 0.95. Does it matter?medium
  25. You have 50,000 measurements from 200 patients. What is your sample size?hard

Mixed effects & clustering

  1. What is a hierarchical Bayesian model?hard

Bayesian methods

  1. What is the posterior predictive distribution?hard
  2. What is the marginal likelihood (evidence) and why is it hard?hard
  3. How do you interpret a Bayes factor?hard
  4. How does Bayesian A/B testing work?medium
  5. How does Metropolis-Hastings work?hard
  6. What is Gibbs sampling?hard
  7. Hamiltonian Monte Carlo — intuition.hard
  8. How do you diagnose MCMC convergence?hard
  9. Burn-in and thinning — why?medium
  10. Variational inference vs MCMC.hard
  11. Derive the ELBO for VI.hard
  12. Informative vs non-informative priors — the tradeoff.medium
  13. WAIC and LOO-CV — Bayesian model comparison.hard
  14. What is a prior predictive check?hard
  15. Posterior predictive check — how do you use it?hard
  16. When should you use a bandit instead of an A/B test?hard

Causal inference basics

  1. Why doesn't correlation imply causation?medium
  2. What is Simpson's paradox?hard
  3. Users who adopt feature X churn less. Can you say the feature reduces churn?easy
  4. What is SUTVA?hard
  5. How do you handle confounders in observational studies?medium
  6. An aggregate trend reverses once you stratify. Which conditioning set is correct?medium
  7. What is a collider and why is conditioning on it bad?hard
  8. What is Pearl's backdoor criterion?hard
  9. When is the frontdoor criterion useful?hard
  10. What is a propensity score?hard
  11. Inverse Probability Weighting — how does it work?hard
  12. Why is a doubly-robust estimator useful?hard
  13. What makes a valid instrumental variable?hard
  14. Difference-in-differences — setup and assumption.hard
  15. How do you defend the parallel-trends assumption?hard
  16. Regression discontinuity design — how does it work?hard
  17. What is synthetic control?hard
  18. Mediation analysis — direct vs indirect effects.hard
  19. How do you estimate heterogeneous treatment effects (HTE)?hard
  20. How does Simpson's paradox strike A/B tests?hard
  21. How do you test in a marketplace / network with SUTVA violations?hard
  22. Quantile treatment effect (QTE) vs ATE.hard
  23. Real interview: your model performs great in A/B but flops post-launch. Why?hard
  24. You cannot randomize. What is the strongest causal claim you can still make?hard

Experimentation & A/B testing

  1. What are the pillars of a solid A/B test design?medium
  2. What is Sample Ratio Mismatch (SRM) and why check it?medium
  3. What are guardrail metrics?easy
  4. OEC — Overall Evaluation Criterion — what is it?hard
  5. Novelty vs primacy effects in long experiments.hard
  6. Switchback experiments — when and how?hard
  7. What is the winner's curse in A/B testing?hard
  8. How do you measure long-term effects when A/B tests are short?hard
  9. How does a company scale to running 1000+ experiments concurrently?hard

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