Statistics is the theoretical backbone of data science and a core topic in virtually every data scientist interview. From understanding p-values to designing A/B tests correctly, statistical thinking separates strong candidates from those who can only run model.fit(). This guide covers the 40 most important statistics interview questions with complete answers, covering probability, distributions, inference, regression, and experimental design.
Probability and Foundations
Q1. What is the difference between probability and statistics?
Probability works forward from a known model to predict outcomes. Given a fair coin (known model), what is the probability of 7 heads in 10 flips? Statistics works backward — from observed data to infer the underlying model. Given that we observed 7 heads in 10 flips, is the coin fair? Probability uses deductive reasoning (from model to data); statistics uses inductive reasoning (from data to model). Both are needed in data science: probability theory provides the mathematical framework, statistics provides the tools to apply it to real observations.
Q2. What is conditional probability? State Bayes’ theorem.
Conditional probability P(A|B) is the probability of event A occurring given that event B has already occurred. P(A|B) = P(A ∩ B) / P(B). Bayes’ theorem relates the conditional and marginal probabilities: P(A|B) = P(B|A) × P(A) / P(B). In plain English: posterior probability is proportional to the likelihood times the prior. Classic example: P(disease|positive test) = P(positive test|disease) × P(disease) / P(positive test). Even a highly accurate test can have low precision when the disease is rare — Bayes’ theorem quantifies exactly how base rates affect test interpretation.
Q3. What is the law of large numbers vs the central limit theorem?
The Law of Large Numbers states that as sample size increases, the sample mean converges to the true population mean. Given enough data, your estimate will be accurate. The Central Limit Theorem (CLT) states that regardless of the population distribution, the distribution of sample means approaches a normal distribution as sample size grows (n > 30 is a common rule of thumb). The CLT is what justifies using normal-distribution-based hypothesis tests (t-tests, z-tests) on non-normal data. They are related but different: LLN is about accuracy of the estimate; CLT is about the shape of the sampling distribution.
Q4. What is the expected value and variance? Give formulas.
Expected value (mean) E[X] = Σ x × P(X=x) for discrete distributions, or ∫ x × f(x) dx for continuous. It is the long-run average value. Variance Var(X) = E[(X – E[X])²] = E[X²] – (E[X])² — measures spread around the mean. Standard deviation is √Var(X), in the same units as the data. Key properties: E[aX + b] = aE[X] + b; Var(aX + b) = a²Var(X); E[X + Y] = E[X] + E[Y] (always); Var(X + Y) = Var(X) + Var(Y) only if X and Y are independent.
Q5. What is covariance and correlation? What is the difference?
Covariance measures the direction of the linear relationship between two variables: Cov(X,Y) = E[(X – μ_X)(Y – μ_Y)]. Positive covariance means they move together; negative means they move opposite. Problem: covariance is in the product of the original units, making it hard to interpret in absolute terms. Correlation standardises covariance: ρ = Cov(X,Y) / (σ_X × σ_Y). Pearson correlation ranges from -1 (perfect negative linear) to +1 (perfect positive linear), with 0 meaning no linear relationship. Correlation is dimensionless and comparable across pairs of variables. Both measure only linear relationships — two variables can have zero correlation but strong non-linear dependence.
Distributions and Statistical Models
Q6. Name five common probability distributions and their real-world applications.
Normal (Gaussian): heights, IQ scores, measurement errors, residuals in regression. The foundation of parametric statistics due to the CLT. Binomial: number of successes in n independent trials (conversion rate testing, defect counting). Poisson: count of rare events in a fixed interval (server errors per hour, customer arrivals per minute, insurance claims). Exponential: time between events in a Poisson process (time between customer arrivals, equipment failure time). Uniform: random sampling without preference (simulation, baseline for random number generation). Beta: models proportions and probabilities, used as a prior in Bayesian A/B testing. Log-normal: products of many small random factors (income, stock prices, city populations).
Q7. What is the difference between the PDF and CDF?
For a continuous random variable, the Probability Density Function (PDF) f(x) gives the relative likelihood of the variable taking a value near x. The PDF itself is not a probability — you must integrate over an interval: P(a ≤ X ≤ b) = ∫[a to b] f(x) dx. The Cumulative Distribution Function (CDF) F(x) = P(X ≤ x) gives the probability that the variable is less than or equal to x. The CDF is the integral of the PDF. The Quantile Function (inverse CDF) gives the value x such that P(X ≤ x) = p — used for confidence intervals and percentile calculations.
Q8. When is the normal distribution assumption violated and what do you do?
Common violations: heavy tails (financial returns have extreme events more often than normal); skewed distributions (income, transaction amounts); multimodal distributions (mixture of populations); discrete data (counts cannot be truly normal). Detection: histogram, Q-Q plot, Shapiro-Wilk test. Remedies: (1) Transform the data — log transform for right-skewed, Box-Cox for general transformations. (2) Use non-parametric tests that do not assume normality — Mann-Whitney U instead of t-test, Kruskal-Wallis instead of ANOVA. (3) Use the CLT — if your sample is large enough (n > 30), the sampling distribution of the mean is approximately normal regardless of population shape, validating t-tests even on non-normal data.
Hypothesis Testing
Q9. Explain Type I and Type II errors and the relationship between α and β.
Type I error (false positive): rejecting H₀ when it is actually true — concluding there is an effect when there is none. The probability of Type I error is α (significance level), which you control by setting the threshold. Typical values: 0.05 or 0.01. Type II error (false negative): failing to reject H₀ when it is false — missing a real effect. The probability of Type II error is β. Statistical power = 1 – β — the probability of correctly detecting a real effect. There is a fundamental trade-off: reducing α (more stringent) increases β for a given sample size. The only way to reduce both simultaneously is to increase sample size. Power analysis before a study determines the sample size needed to detect a given effect size at desired α and power levels.
Q10. What is a p-value? What are common misconceptions?
The p-value is the probability of observing data at least as extreme as the actual data, assuming the null hypothesis is true. It is not: the probability that H₀ is true; the probability that the result is due to chance; the probability that H₁ is true; or a measure of effect size or practical significance. A p-value of 0.03 means “if H₀ is true, there is a 3% chance of seeing this or more extreme data” — not “there is a 3% chance the null is true.” A small p-value indicates that the data is unlikely under H₀, so we reject H₀. It says nothing about how large or practically meaningful the effect is. Always report effect size alongside p-values.
Q11. What is statistical power and what factors affect it?
Statistical power is the probability of correctly rejecting a false null hypothesis (detecting a real effect). Power = 1 – β. Four factors: (1) Effect size — larger real effects are easier to detect; power increases with effect size. (2) Sample size — more data means more power; the most important factor you control. (3) Significance level (α) — higher α (e.g., 0.10 vs 0.05) gives more power but more false positives. (4) Variance — lower variance in the data gives more power. Standard practice: power ≥ 0.80 (80% chance of detecting a true effect). Use power analysis before collecting data to determine the minimum sample size needed.
Q12. When should you use a one-tailed vs two-tailed test?
A two-tailed test checks whether the parameter is significantly different from the null value in either direction. A one-tailed test checks only one direction — whether the parameter is significantly greater than or significantly less than the null value. Use two-tailed by default. Use one-tailed only when: the alternative is genuinely directional before seeing the data (you have strong prior reason to expect only one direction of effect); and the consequences of an effect in the other direction would be the same as no effect. Never switch from two-tailed to one-tailed after seeing the data — that is p-hacking.
Q13. What is multiple testing and how do you correct for it?
When you run multiple hypothesis tests simultaneously, the probability of at least one false positive grows rapidly. With 20 tests at α=0.05, you expect one false positive just by chance even with no real effects. Bonferroni correction: divide α by the number of tests (α/n) — conservative, reduces power. Benjamini-Hochberg (FDR): controls the False Discovery Rate (expected proportion of false positives among rejected hypotheses) rather than the family-wise error rate — less conservative, preferred in exploratory analysis. When to apply: any time you run more than one test on the same data — multiple features in a study, multiple subgroup analyses, multiple time points.
Regression and Experimental Design
Q14. What are the assumptions of linear regression and how do you check them?
The four key assumptions (LINE): Linearity — the relationship between predictors and response is linear. Check: residuals vs fitted plot should show no pattern. Independence — observations are independent of each other. Check: Durbin-Watson test for autocorrelation in residuals. Normality — residuals are normally distributed. Check: Q-Q plot of residuals, Shapiro-Wilk test. Equal variance (homoscedasticity) — residual variance is constant across all fitted values. Check: scale-location plot (standardised residuals vs fitted), Breusch-Pagan test. Violations: Linearity → add polynomial terms or transform variables. Non-normality → bootstrap or GLM. Heteroscedasticity → weighted least squares or log-transform the response. Autocorrelation → include lag features or use time series models.
Q15. What is multicollinearity and why is it a problem in regression?
Multicollinearity occurs when two or more predictors are highly correlated with each other. It is a problem because: it makes individual coefficient estimates unreliable and sensitive to small changes in data; standard errors of coefficients become large, widening confidence intervals; it makes it hard to isolate the individual effect of each correlated feature. Detection: correlation matrix (pairs with |r| > 0.8), Variance Inflation Factor (VIF > 10 indicates severe multicollinearity). Solutions: remove one of the correlated features; combine them into a single feature (sum or average); use PCA to create orthogonal components; use Ridge regression, which is robust to multicollinearity due to L2 regularisation.
Q16. How do you design a proper A/B test?
Step 1: Define the hypothesis and primary metric before any data collection. Step 2: Power analysis — determine sample size needed to detect your minimum meaningful effect at α=0.05 and 80% power. Step 3: Randomise — assign users randomly to control (A) and treatment (B) groups. Randomisation must be independent and at the right unit of analysis. Step 4: Run the experiment for a pre-determined duration — not until you see significance (optional stopping inflates Type I error). Step 5: Analyse — use a two-sample t-test or z-test for means/proportions, check guardrail metrics that should not change. Step 6: Decision — if p-value < α, reject H₀ and consider shipping. Report effect size and confidence interval, not just p-value.
Q17. What is the difference between correlation and causation? How do you establish causation?
Correlation means two variables move together. Causation means one variable directly causes changes in another. Correlation does not imply causation because: both may be driven by a confounding variable (ice cream sales and drowning both increase in summer — neither causes the other); the relationship may be coincidental (spurious correlations); the causation may be reversed. Establishing causation requires: (1) Randomised Controlled Trial (RCT) — the gold standard, randomly assign subjects to treatment/control. (2) Natural experiments — exploit accidental randomisation in real life (lottery assignment, regulatory changes). (3) Causal inference methods — difference-in-differences, instrumental variables, regression discontinuity, or propensity score matching when true randomisation is impossible.
Q18. What is Simpson’s paradox? Give an example.
Simpson’s paradox occurs when a trend appears in several groups of data but disappears or reverses when the groups are combined. Classic example: a hospital has two treatments for kidney stones. Treatment A succeeds 78% overall; Treatment B succeeds 83% overall. So B appears better. But when stratified by stone size: for small stones, A succeeds 93%, B 87%. For large stones, A succeeds 73%, B 69%. A is better for both subgroups! The paradox arises because Treatment B was used more on small stones (easier cases), inflating its overall success rate. Simpson’s paradox is a warning always to examine subgroup analyses and confounders before drawing conclusions from aggregate statistics.
Bayesian Statistics
Q19. What is the difference between frequentist and Bayesian statistics?
Frequentist statistics treats probability as the long-run frequency of events in repeated experiments. Parameters are fixed (unknown) constants; data is random. Confidence intervals and p-values are properties of the procedure, not of any specific outcome. Bayesian statistics treats probability as a degree of belief. Parameters themselves have probability distributions (priors). After observing data, we update beliefs using Bayes’ theorem to get a posterior distribution. Bayesian inference gives direct probability statements about parameters — P(parameter > 0 | data) — which is what people often incorrectly think frequentist p-values say. In practice: Bayesian methods are more flexible (incorporate prior knowledge, handle small samples better) but computationally more demanding.
Q20. What is a prior, likelihood, and posterior in Bayesian inference?
Prior P(θ): your belief about the parameter θ before seeing the data. It encodes prior knowledge or assumptions. A flat/uninformative prior assumes all values equally likely; an informative prior incorporates domain knowledge. Likelihood P(data|θ): how probable is the observed data given a specific value of θ? This is the data’s contribution — it comes from the chosen statistical model. Posterior P(θ|data) ∝ P(data|θ) × P(θ): your updated belief about θ after seeing the data. The posterior combines prior belief with evidence from the data. With enough data, the likelihood dominates and the prior becomes irrelevant — Bayesian and frequentist results converge.
Q21–30 (Rapid fire statistics):
Q21. What is a confidence interval? A 95% CI means: if we repeated the experiment 100 times and computed a CI each time, 95 of those intervals would contain the true parameter. It is a property of the procedure, not the specific interval. It is NOT “95% probability the parameter is in this interval.”
Q22. What is effect size and why report it? Effect size measures the practical magnitude of a difference (Cohen’s d for means, Pearson r for correlation, odds ratio for proportions). A statistically significant result with tiny effect size may have no practical value. Always report alongside p-values.
Q23. What is heteroscedasticity? Non-constant variance of residuals across the range of fitted values. Violates linear regression assumptions, causing inefficient estimates and unreliable standard errors. Fix with weighted least squares or log-transforming the response.
Q24. What is autocorrelation? Correlation of a variable with its own past values. Violates the independence assumption of regression. Common in time series data. Detected with ACF plots or Durbin-Watson test. Handle with ARIMA models or adding lag features.
Q25. What is the difference between parametric and non-parametric tests? Parametric tests (t-test, ANOVA) assume data follows a specific distribution (usually normal). Non-parametric tests (Mann-Whitney, Kruskal-Wallis) make fewer distribution assumptions — use them when normality is violated or sample size is small.
Q26. What is bootstrapping? A resampling technique that repeatedly samples with replacement from the data to estimate the sampling distribution of any statistic. Requires no distributional assumptions. Use to compute confidence intervals for any metric (median, correlation, custom metrics).
Q27. What is regularisation from a statistical perspective? Regularisation is equivalent to placing a prior on model parameters. L2 (Ridge) corresponds to a Gaussian prior; L1 (Lasso) corresponds to a Laplace prior. Both shrink parameter estimates toward zero, reducing variance at the cost of some bias.
Q28. What is the difference between standard deviation and standard error? Standard deviation (SD) measures the spread of individual data points around the mean. Standard error (SE) = SD/√n measures the precision of the sample mean estimate. SE decreases as sample size grows; SD stays roughly constant.
Q29. What is a Q-Q plot used for? A quantile-quantile plot compares the quantiles of your data to the quantiles of a theoretical distribution (usually normal). If the data follows that distribution, points fall on a straight diagonal line. Deviations indicate departures: S-shaped curves indicate different skewness, heavy tails show as points curving away at the ends.
Q30. What is a Z-score? Z = (x – μ) / σ. It measures how many standard deviations a value is from the mean. Allows comparison across different scales. Used for standardisation in preprocessing and for outlier detection (|Z| > 3 is commonly considered an outlier).
Quick Code Reference
from scipy import stats
import numpy as np
# One-sample t-test
t_stat, p = stats.ttest_1samp(data, popmean=50)
# Two-sample t-test (check equal variance first)
_, p_var = stats.levene(group1, group2)
t_stat, p = stats.ttest_ind(group1, group2, equal_var=(p_var > 0.05))
# Effect size: Cohen's d
d = (group1.mean() - group2.mean()) / np.sqrt((group1.std()**2 + group2.std()**2) / 2)
print(f"Cohen's d = {d:.3f} ({'small' if abs(d)<0.5 else 'medium' if abs(d)<0.8 else 'large'})")
# Confidence interval
ci = stats.t.interval(0.95, df=len(data)-1,
loc=np.mean(data), scale=stats.sem(data))
# Bootstrap confidence interval
boot_means = [np.mean(np.random.choice(data, len(data))) for _ in range(10000)]
ci_boot = np.percentile(boot_means, [2.5, 97.5])
Conclusion
Statistics interviews for data science roles increasingly go beyond textbook formulas to test applied judgment — when to use which test, how to design experiments correctly, how to interpret results without common mistakes. The highest-leverage topics are hypothesis testing (especially Type I/II errors, power, and multiple testing), A/B test design (sample size calculation, avoiding peeking), regression assumptions, and the distinction between statistical and practical significance. Study these deeply, practice explaining them without jargon, and you will stand out from candidates who can only recite formulas.
