# The statistics questions quant and data science interviews ask

Source: https://www.techinterview.org/post/3233477272/statistics-questions-quant-data-science-interviews/
Updated: 2026-08-08 · techinterview.org

"Explain what a p-value is." That's often the first real question in a quant research or data science screen, and a lot of strong-looking candidates fumble it. They say it's the probability the null hypothesis is true, or the chance the result happened randomly. Both are wrong, and the interviewer usually knows within a few seconds whether you understand inference or you memorized a definition you can't defend under one follow-up.

A p-value is the probability of observing data at least as extreme as what you saw, computed under the assumption that the null hypothesis holds. The conditioning on the null is the entire idea. A p-value of 0.03 does not mean there's a 3% chance the null is true. It means that if the null were true, data this extreme or more would show up 3% of the time. Flip that around and you're committing the base-rate error the interviewer is fishing for. If they push, they'll ask what happens to your reading if the effect you're testing is very unlikely a priori. The correct answer is that a small p-value combined with a low prior can still leave the alternative unlikely, which is exactly why people who reason only from p-values ship false positives.

## Hypothesis testing and the errors nobody wants to own

Once you're past the definition, the follow-ups probe whether you can reason about the two ways a test fails. A type I error rejects a true null (a false positive), and its rate is the significance level you set, usually 0.05. A type II error fails to reject a false null (a false negative), and its rate is beta. Power is 1 minus beta, the probability you detect a real effect when one exists. Interviewers like to ask what raises power: a larger sample, a bigger true effect, lower variance, or a looser alpha. They're checking whether you see the direct tension between the two error rates. Tighten alpha to catch fewer false positives and you lose power, so you miss more real effects. On a trading desk that tradeoff is not academic. A signal team that sets alpha too loose will chase noise into production and bleed money on strategies that were never real.

A common curveball: you run 100 independent tests at alpha 0.05 and none of the effects are real. How many do you expect to come back significant? Five. That's the multiple-comparisons problem, and if you can name Bonferroni (divide alpha by the number of tests) or the Benjamini-Hochberg procedure for controlling the false discovery rate, you've shown you've actually run experiments rather than read about them.

## Confidence intervals mean something narrower than people think

"You compute a 95% confidence interval and get [2.1, 4.8]. What does the 95% refer to?" The wrong answer, and it's everywhere, is that there's a 95% probability the true parameter lies in that interval. Under the frequentist framing the true parameter is fixed, not random, so it's either in your interval or it isn't. The 95% describes the procedure: if you repeated the sampling many times and built an interval each time, about 95% of those intervals would contain the true value. A Bayesian credible interval is the one that actually supports the "95% probability the parameter is in here" statement, and drawing that distinction cleanly is a strong signal that you know which framework you're standing in.

## Deriving an MLE on the whiteboard

Maximum likelihood is the estimation question that separates people who can turn a model into math from people who name-drop it. A clean version: you flip a coin n times and see k heads. Derive the MLE for the probability of heads.


```
L(p)     = p^k (1-p)^(n-k)
log L(p) = k log p + (n-k) log(1-p)
d/dp     = k/p - (n-k)/(1-p) = 0
=>  p_hat = k / n
```


The estimator is just the sample proportion, which feels obvious, but the interviewer wants the steps: write the likelihood, take the log because sums are easier to differentiate than products, set the derivative to zero, and confirm it's a maximum rather than a minimum. A good follow-up is whether the MLE is biased. For the coin it's unbiased, but MLE in general is not. The textbook counterexample is the variance estimator that divides by n instead of n-1, which underestimates the true variance; the fix, Bessel's correction, is the kind of detail quant researchers are expected to know cold.

## Linear regression, and the assumptions people skip

Regression shows up in nearly every quant and data interview because it's the workhorse for turning features into a prediction, and because it has assumptions that are easy to state and easy to violate. OLS gives you the best linear unbiased estimator when a handful of conditions hold: the relationship is linear in the parameters, the errors have mean zero and constant variance (homoscedasticity), the errors are uncorrelated with each other, and no predictor is a perfect linear combination of the others. Drop normality of the errors and your coefficient estimates are still fine; you just lose the exact small-sample inference for p-values and intervals.

The questions that trip people up target what breaks and how you'd notice. Heteroscedasticity, variance that grows with the fitted value, leaves your coefficients unbiased but wrecks the standard errors, so your p-values lie; you catch it by plotting residuals against fitted values and looking for a fan shape. Multicollinearity, two predictors that move together, inflates the variance of the coefficient estimates, so signs flip and t-stats collapse even when the model predicts well; the variance inflation factor is the standard diagnostic. And R-squared, the number everyone quotes, only tells you the fraction of in-sample variance the model explains. It never goes down when you add a predictor, which is why adjusted R-squared or out-of-sample error is what a serious interviewer wants to see you reach for.

## Frequentist versus Bayesian, with a number attached

Expect at least one Bayes computation, because it's the fastest way to see whether you can update on evidence. The canonical version: a test for a condition is 99% accurate, the condition affects 1 in 10,000 people, someone tests positive, what's the probability they have it? People blurt out 99%. The answer is under 1%, because the base rate is so low that false positives from the huge healthy population swamp the true positives. Write P(condition | positive) = P(positive | condition) P(condition) / P(positive), plug in 0.99 times 0.0001 over (0.99 times 0.0001 plus 0.01 times 0.9999), and you land near 0.98%. The lesson the interviewer is drilling is the same one behind the p-value question: evidence has to be weighed against a prior, and ignoring the base rate produces confident nonsense.

Where the split matters in practice is small data. A quant researcher estimating the win rate of a new signal off 40 trades gets almost nothing stable from the raw frequency, but a Bayesian estimate with a sensible prior shrinks toward something reasonable and quantifies how unsure you should be. Say that out loud and you've shown you treat these as tools you'd pick between, not trivia to recite.

## What different firms push on

The weighting shifts by seat. A quant research role at Two Sigma, D. E. Shaw, or a Jane Street research desk leans hard on distributions, MLE, and reasoning about estimator variance under small samples and fat tails. A data science interview at a large tech company spends more time on A/B testing: how you'd size an experiment, what a sequential test buys you, how to handle a metric that's a ratio, and what to do when the treatment effect is tiny relative to the variance. Both camps ask about the bias-variance tradeoff, because it's the idea that connects statistics to machine learning: a model that's too simple is biased, one that's too flexible has high variance, and the estimator that generalizes minimizes total error out of sample.

A few questions come up often enough to rehearse out loud:

- What's the difference between covariance and correlation, and why do you report correlation?

- Your A/B test crosses significance on day two of a two-week run. Why shouldn't you stop and ship?

- How would you detect and handle outliers in a dataset you've never seen?

- What does the central limit theorem let you assume, and when does it fail you?

The through-line in the ones people miss is that most statistics interview mistakes aren't computational. They're interpretive: claiming a p-value or a confidence interval means something slightly stronger than it does. The table pairs the concepts that get confused with the answer that fails and the one that lands.

| Concept | The answer that fails | What it actually means |
| --- | --- | --- |
| p-value | The probability the null hypothesis is true | P(data at least this extreme, given the null is true) |
| 95% confidence interval | 95% probability the true parameter is inside this interval | Over repeated sampling, 95% of intervals built this way contain the true parameter |
| Statistical power | The chance the result is correct | P(reject the null, given the alternative is true) = 1 minus beta |
| R-squared | How good the model is | Fraction of in-sample variance explained; never decreases as predictors are added |
| Failing to reject the null | The null is true, so there's no effect | Not enough evidence to reject; the test may just be underpowered |
| MLE | Always the unbiased best estimate | The parameter that maximizes the likelihood; consistent, but can be biased in small samples |

If you prepare one thing for these rounds, make it the ability to state what your number does not tell you. Anyone can compute a p-value or fit a regression. The candidate who gets the offer is the one who, when the interviewer leans in and asks what that actually means, answers with the boundary of the claim instead of overselling it.