In this article
Before you start
This page belongs to the quantitative finance interview guide, which covers the wider set of rounds.
Why interviewers ask this
A statistics question in a quant round is rarely about the formula. It is a way to see whether you can say what a number means and what it does not mean. The wording of the question is usually short, and the follow-up is where the real test sits.
A weak answer recites a definition and stops: "a p-value is the probability the null is true." A strong answer gives the correct reading, shows a small calculation, and names the misreading that people fall into. A Staff-level answer also says when the textbook result stops holding, for example when data have heavy tails or when the same data are tested many times.
The topics below follow the standard syllabus of the e-Handbook of Statistical Methods, the MIT OpenCourseWare probability and statistics course and the Penn State regression notes. The same ideas also help on the markets side, as in sales and trading interview questions.
Statistics for quant interviews from first principles
- Base rate
- A base rate is the share of a population that has a condition before any test or evidence is looked at. It is the starting probability in a Bayes calculation, and ignoring it is the classic reason a positive test result is misread as near-certain.
- p-value
- A p-value is the probability of seeing data at least as extreme as the observed data if the null hypothesis were true. It measures surprise under an assumption; it is not the probability that the assumption is true.
In plain words
Think of a weather forecaster. A forecast is a belief; each new reading is evidence; statistics is the rulebook for how far the evidence should move the belief and how much to trust the measuring instrument. The analogy breaks in one place: a forecaster can check tomorrow, while in an interview you often have one sample and must say what it supports.
In one precise sentence, statistical inference uses a sample to say something about a population, together with an honest measure of how wrong that statement could be. The four steps below build that up in order. Each step is short. Each one ends with a number you can check by hand.
Step 1: Bayes' theorem needs the base rate
Bayes' theorem is the rule that converts the probability of a test result given a condition into the probability of the condition given the result, using the base rate; the answer is the posterior, the updated probability after the evidence. Write D for "has the condition" and + for "tests positive":
Input: a base rate, the test's hit rate and its false-alarm rate. Output: the probability the condition is real after a positive result. Take an illustrative condition with a base rate of 1%, a test that detects it 90% of the time and a false-alarm rate of 5%. Out of 10,000 people, 100 have it and 90 of those test positive; of the 9,900 who do not, 495 also test positive. A positive result then means 90 out of 585. The formula gives the same answer, with the numerator 0.9 · 0.01 = 0.009 and the denominator 0.009 + 0.05 · 0.99 = 0.0585:1
The same test with a base rate of 50% gives 0.9474.1 The test did not change; the base rate did.
Step 2: The standard error shrinks with the square root of the sample size
The standard error is the standard deviation divided by the square root of the sample size, which is why a bigger sample helps but with diminishing returns. Quadrupling the sample size only halves the standard error. With σ for the standard deviation of the data and n for the sample size:
Input: a sample mean, a standard deviation and a sample size. Output: how far the mean could plausibly sit from the truth. The central limit theorem (CLT) is the result that the average of many independent, identically distributed draws with finite variance is close to normally distributed, whatever the shape of the raw data. It says the sample mean becomes normal, not that the data do.
With an illustrative standard deviation of 10, taken as known, and a sample of 25, the standard error is 2.0; at a sample of 100 it falls to 1.0.1 You now know how precise an average is; the next step turns that precision into statements.
Step 3: A p-value measures surprise, not the odds the null is true
A p-value is computed assuming the null hypothesis is true. The null hypothesis is the default claim, for example "the true mean is 100" (illustrative). The test asks how surprising the sample would be under that claim.
Input: the sample mean, the claimed mean and the standard error. Output: a z-score, the distance in standard errors, and a p-value from the normal curve. With x̄ for the sample mean and μ₀ for the claimed mean:
With an illustrative sample mean of 102, a claimed mean of 100 and a standard error of 2.0, the z-score is 1.0 and the two-sided p-value is 0.3173: not surprising.1 A sample mean of 104 gives a z-score of 2.0 and a p-value of 0.0455.1
The confidence interval answers the neighbouring question: which claimed means would not be rejected. The interval is , using the standard 1.96 multiplier for 95% from the normal curve, which is valid when the standard deviation is known.1 For a mean of 102 it is 98.08 to 105.92.1 The 95% describes the method over repeated samples, not this one interval.
When the standard deviation is instead estimated from the sample, use the t-distribution with n − 1 = 24 degrees of freedom (the sample size minus the one mean already estimated). Its 95% multiplier is about 2.064, from the t-table of the NIST e-Handbook, so the interval widens to 97.87 to 106.13.1 The e-Handbook of Statistical Methods covers both tools.
Step 4: Regression fits a line, and the residuals tell you whether to trust it
Ordinary least squares (OLS) is the method that fits the straight line minimising the sum of squared vertical gaps between the data and the line. The gaps are the residuals. Regression assumptions are checked on the residuals: a patternless scatter supports the fit, while a curve or a fan shape does not. Fitting a line this way is also the simplest case of the training idea in deep learning.
Input: pairs of values. Output: a slope, an intercept and a measure of fit called R-squared, the share of the variation in the outcome that the line explains. The OLS slope is the covariation of x and y divided by the variation of x:
For the five illustrative points of the worked example this is 6 / 10 = 0.6.1 Correlation is a measure of how closely two variables follow a straight-line pattern, running from minus one to one. The Penn State regression notes list the usual assumptions: a linear relationship, independent errors, constant variance and roughly normal errors.
Step-by-step description
- A test result and a base rate give a posterior.
- The estimate's wobble shrinks with root n.
- Wobble becomes an interval and a p-value.
- A fitted line is checked on its residuals.
- Bayes' theorem turns a test result into a posterior probability.
- The standard error is the standard deviation divided by the square root of the sample size.
- A p-value is computed assuming the null hypothesis is true.
- Regression assumptions are checked on the residuals.
Knowing it is half the job. Saying it is the other half.
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Every number below is computed or illustrative, so you can redo it with a calculator. The Bayes counts use an illustrative 10,000 people, a base rate of 1%, a hit rate of 90% and a false-alarm rate of 5%.1
| Quantity | Value |
|---|---|
| People with the condition, per 10,000 | 100 |
| True positives | 90 |
| False positives | 495 |
| All positives | 585 |
| Posterior at a base rate of 1% | 0.1538 |
| Posterior at a base rate of 50% | 0.9474 |
| Standard error, sample of 25, standard deviation 10 | 2.0 |
| 95% interval around a sample mean of 102 | 98.08 to 105.92 |
| Two-sided p-value at a sample mean of 102 | 0.3173 |
| Two-sided p-value at a sample mean of 104 | 0.0455 |
Of 585 positive results, 90 come from people who have the condition, and the other 495 are false positives.1 The false positives win because the healthy group is far larger.
Text description of the diagram
- Of 585 positive results, 90 come from people who have the condition.
- The other 495 are false positives.
The regression uses five illustrative points: x values 1 to 5 and y values 2, 4, 5, 4, 5. The fitted line has slope 0.6 and intercept 2.2. The residuals are minus 0.8, 0.6, 1.0, minus 0.6 and minus 0.2, and R-squared is 1 − 2.4 / 6 = 0.6, where 2.4 is the sum of squared residuals and 6 is the total squared variation of y around its mean.1
Trade-offs and edge cases
The textbook results hold under conditions, and interviewers like to find the edge. Four cases matter.
Heavy tails. The CLT needs finite variance. With infinite variance the sample mean does not become normal at all, and with a very large but finite variance it converges so slowly that a normal interval can be too narrow, as can happen for returns with rare very large moves.
Many tests. If you test many hypotheses on the same data, some will give small p-values by chance. A small p-value after a long search is weak evidence unless the search is accounted for, for example by a multiple-testing correction.
Zero correlation is not independence. Take x equal to minus one, zero or one with equal probability, and y equal to x squared. The correlation is 0.0, yet y is fully determined by x.1 Correlation only measures straight-line association.
A good fit is not a correct model. A high R-squared does not show the model is right; the residuals can still curve or fan out. A high R-squared also does not imply causation.
Text description of the diagram
- A p-value is the probability of data this extreme if the null is true, not the chance the null is true.
- The central limit theorem makes the sample mean normal, not the data.
The frequentist reading, used above, treats the parameter as fixed and the data as random. A Bayesian reading treats the parameter as uncertain and updates a prior belief, as in Step 1. A strong answer says which reading it is using.
How to answer it in an interview
What is a p-value?
A p-value is the probability of seeing data at least as extreme as mine if the null hypothesis were true. It is not the probability that the null is true; that needs a prior and Bayes' theorem. With illustrative numbers, a sample mean of 104 against a claimed 100 and a standard error of 2 gives a z-score of 2 and a two-sided p-value of about 0.0455. I would add that a small p-value says nothing about the size of the effect, and that testing many hypotheses will produce small p-values by chance, so I would correct for multiple tests.
The spoken structure: give the definition, state what it is not, show one small calculation, then name the limit.
Common mistakes
- "A p-value is the probability the null hypothesis is true." (wrong) That reverses the conditional. (right) "It is the probability of data this extreme if the null were true."
- "A 95% interval has a 95% chance of containing the truth." (wrong) The truth is fixed; the interval is what varies. (right) "The method covers the truth in 95% of repeated samples."
- "The central limit theorem makes the data normal." (wrong) The raw data keep their shape. (right) "The sample mean becomes approximately normal, if the variance is finite."
- "A test with a 90% hit rate means a positive result is 90% likely to be real." (wrong) That drops the base rate. (right) "I need the base rate; here the posterior is 0.1538."
- "Zero correlation means independent." (wrong) A curve can have zero correlation. (right) "Correlation only measures straight-line association."
Follow-up questions
What are the usual statistics interview questions for quant roles?
The topics are Bayes' theorem and conditional probability, expectation and variance, the central limit theorem, confidence intervals, hypothesis tests and regression. Each shows up as a short definition question followed by a "what if" about assumptions. Four typical examples, with one-line answers:
- What is the expected value of a fair die roll? The mean of 1 to 6 is 3.5, and the variance is 2.9167.1
- What is the variance of a sum? Var(X+Y) = Var X + Var Y + 2 Cov(X, Y). For two independent dice the covariance is zero and the variance is 5.8333; for X plus itself it is 11.6667, because the covariance term equals Var X.1
- What does maximum likelihood estimation pick? It picks the parameter value that makes the observed data most probable. For 7 heads in 10 tosses of a coin, the estimate of the heads probability is 0.7 (illustrative).1
- A test is 90% sensitive and the base rate is 1%; how likely is a positive to be real? About 15%, as in Step 1.
The quantitative finance guide places them next to probability and market questions.
How should I practise quant statistics interview questions?
Work each topic from a count table or a five-point dataset before using formulas, as in the worked example. Then say the answer aloud in four sentences: definition, what it is not, one number, one limit. For the probability side of the round, work through the Markov and random-walk questions, and browse the wider set of interview guides.
What is a p-value, and what is it not?
A p-value is the probability of data at least as extreme as the observed data if the null hypothesis were true. It is not the probability that the null is true, and it does not measure the size of an effect.
How do you explain a confidence interval?
Say that the 95% belongs to the method: repeated samples would produce intervals that cover the true value in 95% of cases. Then give one interval, such as 98.08 to 105.92, and note that this particular interval either contains the truth or does not.
Does zero correlation mean independence?
No. Correlation only picks up straight-line association, and the example of y equal to x squared has correlation 0.0 although y depends entirely on x. The reverse does hold: independent variables with finite variances have zero correlation.
What are the ordinary least squares assumptions?
The usual list is a linear relationship, independent errors, constant error variance and roughly normal errors, checked on the residuals. A fan-shaped residual plot signals non-constant variance, so the standard errors of the slope cannot be trusted. For how this connects to a full quant loop, see the quantitative finance interview guide.
- Bayes' theorem needs the base rate; a 90% hit rate and a 1% base rate give a posterior of 0.1538.
- The standard error is the standard deviation over the square root of n, so quadrupling the sample halves it.
- A p-value measures surprise if the null is true, and a 95% interval describes a method, not one range.
- The central limit theorem makes the sample mean normal, not the data.
- Zero correlation does not mean independence, and a high R-squared does not mean a correct model.
Sources
- e-Handbook of Statistical Methods — NIST/SEMATECH (US government), accessed 2026-10-26, 26 Oct 2026
- 18.05 Introduction to Probability and Statistics (Spring 2022) — MIT OpenCourseWare, accessed 2026-10-26, 26 Oct 2026
- STAT 501 Regression Methods — Penn State Eberly College of Science, accessed 2026-10-26, 26 Oct 2026
