25 original statistics interview questions with explanations on estimation, likelihood, inference, regression, time series and financial validation.
A backtest can look persuasive because uncertainty was measured incorrectly or because model selection reused the evaluation data. These questions connect estimation and inference to the checks a quant researcher needs in practice. Focus on what is random, which assumptions justify the uncertainty calculation and what the reported statistic does—and does not—establish.
Distinguish a population parameter, an estimator and its realised estimate. State whether the question concerns bias, variance, mean squared error or predictive performance.
Independence, exogeneity and stationarity are substantive assumptions. Explain how serial dependence or heteroskedasticity changes the appropriate uncertainty estimate.
Testing many features or strategies changes the interpretation of a promising result. Keep model selection separate from the final evaluation and report the search process.
Try each question before reading its answer. These original teaching examples remain free to read on this page.
Compare precision and bias using the quantity the decision actually cares about.
Independent identically distributed observations have variance 9. What is the variance of their sample mean for n = 100?
Answer: 0.09.
Explanation: Variances add for independent observations, then averaging divides by n². Standard error is the square root, 0.3.
Two observations each have variance 1 and correlation 0.5. Find variance of their average.
Answer: 0.75.
Explanation: Include the covariance term: (1 + 1 + 2 × 0.5)/4. Treating correlated observations as independent understates uncertainty.
Two estimators are unbiased with variances 1 and 9. Which has lower mean squared error?
Answer: The one with variance 1.
Explanation: Mean squared error equals variance plus squared bias. Unbiasedness alone does not compare an estimator's dispersion.
Estimator A has bias 2 and variance 1; B has bias zero and variance 6. Which has lower MSE?
Answer: A, with MSE 5 versus 6.
Explanation: Square the bias before adding variance. A biased estimator can have better squared-error performance.
Why does sample variance use denominator n − 1 when estimating an IID population variance with the sample mean fitted?
Answer: Estimating the mean removes one degree of freedom.
Explanation: The expected residual sum of squares is (n − 1)σ². Dividing by n would bias this estimator downward.
Derive simple estimates and interpret interval coverage without turning it into a posterior claim.
In 10 independent Bernoulli trials, 7 succeed. What is the maximum-likelihood estimate of p?
Answer: 0.7.
Explanation: Maximise p⁷(1 − p)³ or its log. The likelihood score sets the estimated success rate to the sample proportion.
Independent positive waiting times are exponential with rate λ. Express its MLE using sample mean x̄.
Answer: 1/x̄.
Explanation: The log-likelihood is n log λ − λΣx. Its derivative vanishes at n/Σx; this estimator is not generally unbiased.
Independent observations are normal with known variance. What is the MLE of their common mean?
Answer: The sample mean.
Explanation: Maximising the normal likelihood minimises squared deviations from the candidate mean. Differentiate the quadratic objective.
What does 95% frequentist coverage mean?
Answer: The interval construction contains the fixed true parameter in 95% of repeated samples under its assumptions.
Explanation: It does not assign a 95% posterior probability to the parameter after this one interval has been observed.
A normal sample mean is 10, population standard deviation 2 and n = 100. Give the usual 95% interval.
Answer: Approximately [9.608, 10.392].
Explanation: Use 10 ± 1.96 × 2/√100. Known population variance justifies the normal critical value here.
Keep test error rates and regression coefficient scales separate from informal notions of significance.
What does a p-value of 0.03 mean for a specified test?
Answer: Under the null model, the probability of a test statistic at least as extreme is 3%.
Explanation: It is not the probability the null is true or the probability the result occurred by chance without specifying a model.
What is statistical power?
Answer: The probability of rejecting the null under a specified alternative.
Explanation: Power depends on effect size, noise, sample size and rejection threshold. There is no single power number without an alternative.
Test 20 independent true nulls at a 5% significance level. Find probability of at least one false rejection.
Answer: 1 − 0.95²⁰, approximately 64.2%.
Explanation: Complement the probability that all tests avoid rejection. A per-test error rate is not a family-wide error rate.
To control family-wise error at 5% across 20 tests using Bonferroni, what per-test level should you use?
Answer: 0.0025.
Explanation: Divide 0.05 by 20. The union bound provides control without requiring independence, though the method can be conservative.
In simple regression with an intercept, express the population least-squares slope in terms of covariance and variance.
Answer: Cov(X,Y)/Var(X), when Var(X) > 0.
Explanation: The normal equations yield this ratio. Correlation alone is dimensionless and does not supply the slope's scale.
Training fit, causal interpretation and reliable standard errors require different arguments.
A fitted simple regression has slope 2, mean X = 3 and mean Y = 10. Find its intercept.
Answer: 4.
Explanation: The fitted line passes through sample means: intercept = 10 − 2 × 3. This identity assumes an intercept was included.
A relevant omitted variable is correlated with an included regressor. Can the included coefficient absorb its effect?
Answer: Yes.
Explanation: The residual then contains a component correlated with the regressor. The sign and size depend on both the omitted effect and correlation.
Does a strong observational correlation establish that X causes Y?
Answer: No.
Explanation: Confounding, reverse causation and selection can produce correlation. A causal claim needs an identification argument beyond predictive fit.
Can in-sample R² decrease when adding a regressor to ordinary least squares with the same data and an intercept?
Answer: No.
Explanation: The larger model can set the new coefficient to zero and reproduce the old fit. Improved training fit need not improve prediction.
If errors have mean zero conditional on regressors but nonconstant variance, must OLS coefficients become biased?
Answer: No, under the other standard exogeneity conditions.
Explanation: Heteroskedasticity changes efficiency and usual standard-error formulas. It is not the same violation as correlation between regressors and errors.
Temporal structure changes both estimation uncertainty and the design of a credible evaluation.
Is independently resampling daily returns always an appropriate bootstrap for a dependent return series?
Answer: No.
Explanation: IID resampling destroys dependence. Consider a justified block or model-based resampling scheme and test sensitivity to its assumptions.
Xₜ = 2 + 0.5Xₜ₋₁ + εₜ, with zero-mean innovations and stationarity. Find its mean.
Answer: 4.
Explanation: Take expectations and solve m = 2 + 0.5m. Stationarity makes the time-indexed means equal.
Xₜ = 0.5Xₜ₋₁ + εₜ, with independent innovation variance 3. Find stationary variance.
Answer: 4.
Explanation: Solve v = 0.25v + 3. Independence of the new innovation from the past removes the covariance term.
Why can the sample median be more robust to an extreme outlier than the sample mean?
Answer: An outlier can move the mean arbitrarily, while its magnitude alone does not similarly move the middle order statistic.
Explanation: Robustness concerns sensitivity to contamination, not universal efficiency. The best estimator depends on the distribution and objective.
Why should a financial forecasting model usually be evaluated on later observations than its training data?
Answer: Its operational task is predicting future outcomes from information already available.
Explanation: Random splitting can leak overlapping labels or future regimes. Respect feature availability and use suitable gaps where labels overlap.
Each question has its own page with a short answer and a credit to its source. Hints and full worked solutions are in the question bank with a free account.
Our bank summary dated contains 101 Statistics entries in the bank. Counts describe entries in our curated practice bank, including its source and classification choices. They do not measure employer interview frequency, candidate success rates or how often any one question is asked.
The public examples above were written for these guides. They illustrate the techniques and are not a reproduction or ranking of the bank's sourced questions. The dashboard's broader bank combines sourced questions, worked answers where available and references for book entries without a reproduced solution.
| Subtopic | Bank entries |
|---|---|
| Estimation | 36 |
| Linear Regression | 17 |
| Distributions | 12 |
| Continuous Random Variables | 9 |
| Covariance & Correlation | 6 |
| Expected Value | 4 |
| Hypothesis Testing | 4 |
| Discrete Random Variables | 3 |
| Conditional Expectation | 3 |
| Time Series | 3 |
| Conditional Probability | 2 |
| Events | 1 |
| Bayesian Inference | 1 |
Bank difficulty labels: 69 easy · 26 medium · 6 hard.
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