Statistics interview questions: the complete guide for ML and data roles

In this article

1. The words this guide uses
2. Start here: the learning path
3. Why does the average of many draws look normal?
4. What does a p-value mean, and what does it not mean?
5. How does Bayes' theorem work on a real test?
6. What is P(Y > 3X) for independent standard normals?
7. The path, topic by topic
8. Questions interviewers ask on probability and statistics
9. All 23 guides in Probability and statistics
10. Follow-up questions
11. Sources

The words this guide uses

Statistics interview question

A statistics interview question is a probability or inference problem set in a hiring round, such as a puzzle, a test interpretation or an experiment design, that checks whether the candidate reasons correctly about uncertain data and can say so plainly.

Sampling distribution

A sampling distribution is the distribution of a statistic, such as the sample mean, over many repeated samples of the same size, which tells you how much that statistic would vary from sample to sample.

Bayes' theorem

Bayes' theorem is the rule that updates the probability of a hypothesis after seeing evidence, by multiplying its prior probability by how likely the evidence is under it and rescaling.

iid

iid means independent and identically distributed: each draw comes from the same distribution and no draw influences another, like repeated rolls of one fair die.

Start here: the learning path

The linear algebra page and the distributions page do not depend on each other, so start with either; sampling, tests, estimation, experiments and information theory build on the distributions page, with puzzles alongside. The summaries below describe each topic, and the sections after them link the foundation pages. The same ideas show up in model evaluation, where precision and recall are conditional probabilities estimated from a model's predictions.

1. Linear algebra for machine learning: vectors, eigenvalues and SVD explained
2. What is a Gaussian distribution?
3. What is a t test, and how do you run one?
4. A/B testing interview questions on sample size, p-values, peeking and CUPED
5. How do you estimate pi with a Monte Carlo simulation?
6. Probability interview questions on dice, coins, conditioning and waiting times
7. What are entropy, cross-entropy and KL divergence?
8. MLE vs MAP
9. Bayes theorem interview questions on medical tests, puzzles and naive Bayes
10. How would you design an A/B test for a new ranking algorithm?
11. How do you design an experimentation platform?
12. What does your confidence level mean when building a confidence interval?
13. Why use a t-statistic?
14. Is the importance sampling ratio also called the policy or likelihood ratio?
15. How much should you be willing to pay to play the following game?

What the path covers

Beyond the three foundations, the topics fall into five groups on the map in the next section: t-tests, A/B testing, Monte Carlo, puzzles and basics, and likelihood with Bayes. Mean, correlation and Chebyshev's inequality sit with the puzzles, and Bayes questions sit with maximum likelihood.

The t-test and confidence intervals. The t-test checks whether a mean differs from a reference or from another group when the standard deviation is estimated from the data, and a confidence interval gives the range of means the data are consistent with. Interviewers ask for the t statistic, its assumptions and what the p-value does and does not say.

A/B testing and experiment design. An A/B test randomly splits users into two groups and compares a metric between them. The questions cover design and sample size, reading the result, trust checks such as sample ratio mismatch and the novelty effect, and variance reduction. They build on the t-test and sampling sections here.

Monte Carlo estimation. Monte Carlo methods estimate a quantity by random sampling, such as estimating π from random points in a square. The share of random points that land inside a quarter circle estimates π divided by 4, and the error shrinks like one over the square root of the number of samples, the same law as the spread table below.

Expected value and probability puzzles. The classic questions use dice, coins, birthdays, conditional probability, expectation and variance, the broken stick and Bertrand's chord. The method is to define the random variable, write the expectation and check the extremes.

Maximum likelihood against maximum a posteriori. Maximum likelihood estimation (MLE) chooses the parameters that make the data most likely; maximum a posteriori (MAP) adds a prior belief and chooses the most probable parameters given the data. With a Gaussian prior on a model's weights, MAP is maximum likelihood plus a squared-weight penalty, which is why the prior acts as a regulariser.

Bayes' theorem questions. Beyond the medical-test example below, interviewers ask about priors, Monty Hall, Beta-Binomial updates and naive Bayes, and how to explain Bayes aloud.

Mean against median, correlation against causation, and Chebyshev's inequality. The median is the better summary of skewed data, because a few extreme values pull the mean but not the median. A hidden variable can make two quantities move together without either causing the other. Chebyshev's inequality bounds the share of any distribution beyond k standard deviations of its mean by 1/k squared.

Why does the average of many draws look normal?

The average of many independent draws looks normal because of the central limit theorem: as the sample size grows, the sampling distribution of the mean becomes approximately normal, whatever the shape of the original variable, centred on the population mean with a standard deviation $\sigma /\sqrt{n}$, where $\sigma$ is the standard deviation of the original variable, defined on the Gaussian distribution page (National Institute of Standards and Technology (NIST) and Semiconductor Manufacturing Technology (SEMATECH) e-Handbook, normal distribution). The theorem needs independent draws with a finite variance.

Two facts are easy to blur, so keep them apart. The first is exact at every sample size: the spread of the average of n iid draws is $\sigma /\sqrt{n}$, normal or not. The second is the actual theorem and holds for large n: the shape of the average approaches the normal curve. We checked both with a simulation of a skewed exponential distribution with mean 1 and standard deviation 1, using 20,000 repeats for each sample size.¹ The spread of the average was 1.001, 0.447, 0.182 and 0.100 for n of 1, 5, 30 and 100, against theory of 1.000, 0.447, 0.183 and 0.100. Ten times the data shrinks the spread about threefold.

Text description of the diagram

• The simulated spread of the mean was 1.001, 0.447, 0.182 and 0.100 for sample sizes 1, 5, 30 and 100.
• Theory gives 1.000, 0.447, 0.183 and 0.100, one over the square root of n.

The shape is what the spread alone does not show. Skewness measures lopsidedness and is 0 for a normal curve. The exponential distribution has skewness 2, and the average of n draws has skewness $\frac{2}{\sqrt{n}}$.¹ In the simulation the skewness of the average was 2.010, 0.876, 0.348 and 0.194, against theory of 2.000, 0.894, 0.365 and 0.200. The lopsidedness fades as n grows, and that fading is the move towards the normal curve.

Text description of the diagram

• The simulated skewness of the mean was 2.010, 0.876, 0.348 and 0.194 for sample sizes 1, 5, 30 and 100.
• Theory gives 2.000, 0.894, 0.365 and 0.200, which is 2 over the square root of n.

What does a p-value mean, and what does it not mean?

A p-value is the probability, under a specified model, of a result at least as extreme as the one observed. The American Statistical Association's 2016 statement says p-values can indicate how incompatible the data are with a specified model, and that by itself a p-value does not provide a good measure of evidence about a model or hypothesis (Wasserstein and Lazar, The American Statistician, 2016; American Statistical Association release, March 2016). It is explicit that a p-value is not the probability that the hypothesis is true or that the data arose by chance alone, that decisions should not rest only on whether it passes a threshold, and that it does not measure the size or importance of an effect.

The test behind many interview questions is the t-test: when the standard deviation is unknown, the test uses a t-statistic built from the sample mean, sample standard deviation and sample size, with N minus one degrees of freedom. NIST's worked example has ten particle counts, a t of 1.782 against a critical value of 2.262, so the null hypothesis is not rejected (NIST/SEMATECH e-Handbook, section 7.2.2). In an interview, say what the number does and does not tell you, and pair it with an effect size and an interval.

How does Bayes' theorem work on a real test?

Bayes' theorem corrects the common mistake of answering a test question with the test's accuracy instead of the probability given a positive result. Take illustrative numbers: a condition affects one in a hundred people, a test catches ninety percent of cases and wrongly flags nine percent of healthy people. In ten thousand people, 100 have the condition and 90 test positive correctly; of 9,900 healthy people, 891 test positive wrongly. The chance of having the condition after a positive result is 90 out of 981, or 0.0917, about nine percent.¹

In symbols, with $H$ the condition and $E$ the positive result:

Here $P(E)$ is 981 out of 10,000 in the illustrative counts, the positive results among sick and healthy people alike.

In the interview, count people, not percentages: natural counts keep the base rate in view.

What is P(Y > 3X) for independent standard normals?

For independent standard normal variables X and Y (mean 0, standard deviation 1), P(Y > 3X) is exactly one half, because the pair is symmetric around the origin and the line y = 3x splits the plane into two equal halves. The symmetry needs the means to satisfy ${\mu }_Y=3{\mu }_X$, because $Y-3X$ is normal with mean ${\mu }_Y-3{\mu }_X$ and the probability of a positive value is one half only when that mean is zero.

With both means at 0 it holds; for X and Y iid normal with mean 1 and standard deviation 1, Y minus 3X is normal with mean −2 and variance 10, so the probability is 0.2635, not one half.¹ The random variables and distributions page works the full geometry. Say the symmetry and its assumption out loud.

The path, topic by topic

Each section below summarises one foundation topic and links it; the later topics are described under What the path covers, above.

Text description of the diagram

• Three groups are foundations: distributions, linear algebra for machine learning, and information theory.
• Five groups build on them: t-tests, A/B testing, Monte Carlo, puzzles and basics, and likelihood with Bayes.

Random variables and distributions

A random variable is a number whose value depends on chance, and a distribution lists how likely each value is; the normal distribution is the most common one. The NIST handbook describes the normal curve by its location μ and scale σ, with mean, median and mode all equal to μ and skewness zero (NIST/SEMATECH e-Handbook).

The page covers the two parameters, the central limit theorem and its conditions, z-scores, the symmetry argument for sums of Gaussians, how to build correlated Gaussians from independent ones, and which other families (exponential, Poisson, log-normal) fit which kinds of quantity. Read it first or right after linear algebra: sampling, tests, estimation and information theory all build on it, and it holds the full worked answer to the Y greater than 3X puzzle.

Linear algebra for machine learning

Linear algebra is the language of vectors and matrices in which models and data are written, and the linear algebra for machine learning page covers the parts interviews use: dot products and norms, matrices as transformations, eigenvectors, the singular value decomposition, and low-rank approximation with principal component analysis. It belongs on this path because covariance matrices, regression and principal components are statistics written in matrix form. Read it before regression; it does not need the distributions page. Interviewers use it to check whether you can move between the picture of a vector and the arithmetic of a matrix, and whether you know what an eigenvalue means for a covariance matrix.

Entropy, cross-entropy and Kullback-Leibler (KL) divergence

Entropy measures the average surprise of a distribution, cross-entropy measures the surprise when you use a wrong model, and Kullback-Leibler (KL) divergence is the gap between them. The entropy, cross-entropy and KL divergence page builds each from its definition, shows why KL divergence is not symmetric and so is not a distance, explains why training a classifier on cross-entropy is maximum likelihood and minimises the KL divergence from the data, and ends with perplexity, the exponential of the cross-entropy per token. It follows the distributions page because it needs probabilities over outcomes, and it is the bridge from statistics to the loss functions of machine learning, starting with cross-entropy loss.

Questions interviewers ask on probability and statistics

Expect these five probes: What is random about a random variable? What does a p-value mean? How would you design an A/B test? What is the probability that a positive test is right? How do you find the distribution of the data? The pattern in every case is to define the quantity, state the assumption, then compute.

Follow-up questions

What statistics questions are asked in a data scientist interview?

Expect distributions and the central limit theorem, hypothesis tests and confidence intervals, A/B test design, Bayes' theorem with a base rate, and probability puzzles; the random variables and distributions page is the place to start.

Which statistics questions suit a data analyst interview?

Analyst rounds lean towards summarising data (mean, median, spread), sampling, confidence intervals and reading an A/B test result, so the t-test and A/B testing topics matter most, with the distributions page as the base; a round that also tests a dashboard tool adds questions like the Tableau developer interview questions.

What are the basic statistics interview questions?

The basics are the mean, median and standard deviation, the normal curve, what a p-value is, correlation against causation and the meaning of a confidence level; each is a section on this page or a topic on this path.

What is random about a random variable?

A random variable is a numerical outcome of a chance process, so the randomness is in which value turns up, not in the variable's rule. Once you fix the distribution, every property, such as mean and variance, is a fixed number.

How do you check which distribution the data follows?

Plot the data against the candidate with a histogram and a quantile plot, then test fit with a goodness-of-fit test, and say that no test proves a distribution, it only fails to reject it. The NIST handbook lists estimating a distribution's parameters as a separate step from choosing the family (NIST/SEMATECH e-Handbook, probability distributions).

Sources

1. 1.3.6.6.1. Normal distribution (NIST/SEMATECH e-Handbook of Statistical Methods) — NIST (U.S. National Institute of Standards and Technology), 24 Oct 2026
2. 1.3.6. Probability distributions (NIST/SEMATECH e-Handbook of Statistical Methods) — NIST (U.S. National Institute of Standards and Technology), 24 Oct 2026
3. 7.2.2. Are the data consistent with the assumed process mean? (NIST/SEMATECH e-Handbook of Statistical Methods) — NIST (U.S. National Institute of Standards and Technology), 24 Oct 2026
4. Wasserstein and Lazar (2016), The ASA statement on p-values: context, process, and purpose, The American Statistician, doi:10.1080/00031305.2016.1154108 — American Statistical Association, 7 Mar 2016
5. American Statistical Association releases statement on statistical significance and p-values — American Statistical Association, 7 Mar 2016