Statistics for Quant Interviews: The Complete Prep Guide

Statistics separates quant trading interviews from software engineering interviews. A Google interview tests whether you can implement a red-black tree. A Two Sigma interview tests whether you can tell the difference between a real signal and noise — and that requires statistics.

This guide covers the statistical concepts that appear in every quant research interview, with the depth required to actually impress the interviewers.

1. Distributions You Must Know Cold

Normal Distribution

The bell curve. ~68% of data within 1 standard deviation, ~95% within 2σ, ~99.7% within 3σ.

Z-score: How many standard deviations from the mean? z = (x − μ)/σ.

In trading: Daily stock returns are approximately normal (but with fat tails). This is the Black-Scholes assumption.

Key fact: If X ~ N(μ, σ²), then aX + b ~ N(aμ + b, a²σ²). Linear combinations of normals are normal.

Interview question: "A stock returns 0.1% per day with volatility 1%/day. What's the probability of a daily return worse than −2%?"

z = (−2% − 0.1%) / 1% = −2.1. P(Z < −2.1) ≈ 1.8%.

Lognormal Distribution

If log(X) ~ N(μ, σ²), then X is lognormal. Used for stock prices (prices can't go negative; log returns are approximately normal).

Key fact: If a stock starts at S₀ and daily log returns are N(μ, σ²), then after T days: S_T = S₀ × e^(μT + σ√T × Z) where Z ~ N(0,1).

Common confusion: The EV of a lognormal is e^(μ + σ²/2), not e^μ. The extra σ²/2 term is the "Jensen's gap" — it's why the expected price is higher than e^(μT) × S₀.

Binomial Distribution

X ~ Bin(n, p): count of successes in n independent trials with success probability p.

E[X] = np. Var[X] = np(1−p).

In trading: Win/loss record. If a strategy has a 55% win rate and you take 100 trades, what's the distribution of wins?

X ~ Bin(100, 0.55). E[X] = 55. SD = √(100 × 0.55 × 0.45) = √24.75 ≈ 4.97.

Poisson Distribution

X ~ Poisson(λ): count of rare events in a fixed interval, where λ = expected count.

E[X] = λ. Var[X] = λ. P(X = k) = e^(-λ) × λ^k / k!

In trading: Order arrivals (high-frequency), rare events (market crashes in a year), trade counts.

Interview question: "On average, 3 large trades hit your book per minute. What's the probability of 0 in a given minute?"

P(X = 0) = e^(-3) × 3⁰ / 0! = e^(-3) ≈ 5%

Geometric Distribution

X = number of trials until first success, with P(success) = p.

P(X = k) = (1−p)^(k-1) × p. E[X] = 1/p.

In trading: Time to first fill at a limit price, streaks in win/loss sequences.

Exponential Distribution

Time between events in a Poisson process. If arrivals are Poisson(λ), inter-arrival times are Exponential(λ).

E[X] = 1/λ. The memoryless property: P(X > s+t | X > s) = P(X > t).

This is the continuous analog of the geometric distribution.

2. Hypothesis Testing

This is where most candidates underperform. Hypothesis testing is not just about calculating a p-value — it's a framework for making decisions under uncertainty.

The Framework

1. State H₀ (null hypothesis): the "boring" hypothesis you're trying to disprove.
2. State H₁ (alternative): what you believe is actually true.
3. Choose α (significance level): the false positive rate you'll accept. Usually 0.05.
4. Collect data, compute test statistic.
5. Get p-value: the probability of observing results at least this extreme under H₀.
6. Decision rule: reject H₀ if p < α.

The t-test (One Sample)

Test whether the mean of a sample differs from a hypothesized value μ₀.

Test statistic: t = (x̄ − μ₀) / (s/√n), where s is the sample standard deviation.

Under H₀, this follows a t-distribution with n−1 degrees of freedom.

Trading application: You have a strategy with 50 trades and a mean return of 0.3% per trade, with standard deviation 1.5%. Is this significantly positive?

t = (0.3% − 0%) / (1.5% / √50) = 0.3% / 0.212% ≈ 1.41.

For n−1 = 49 degrees of freedom, t = 1.41 corresponds to p ≈ 0.165. Not significant at the 5% level. You'd need more trades.

Type I and Type II Errors

	H₀ True	H₀ False
Reject H₀	Type I error (α)	Correct ✓
Fail to Reject	Correct ✓	Type II error (β)

Power = 1 − β: the probability of correctly detecting a true effect.

Power depends on:

• Effect size (larger signal = easier to detect)
• Sample size (more data = more power)
• α (higher α = higher power, but more false positives)

For trading: Power is critical. If your strategy has a true Sharpe of 0.5 (modest alpha), you might need 3+ years of daily data to detect it with 80% power.

The Multiple Testing Problem

This is the most important practical concept for quant research.

If you test 100 strategies at α = 0.05, you expect 5 false positives even if no strategy has real alpha. The 5 "significant" results are likely all noise.

Solutions:

Bonferroni correction: Require p < α/n for n tests. For 100 tests, require p < 0.0005. Very conservative.

Benjamini-Hochberg (FDR control): Less conservative. Controls the expected proportion of false positives among all rejected nulls.

Out-of-sample testing: The gold standard. Test hypotheses on data you haven't seen. A strategy that "works" in-sample but fails out-of-sample is just fitting noise.

Interview answer: When asked "how would you test whether a strategy has alpha?" — always mention the multiple testing problem and out-of-sample validation.

3. Regression Analysis

Linear Regression

Y = β₀ + β₁X + ε, where ε ~ N(0, σ²) is the error term.

β₁ (the slope) = Cov(X,Y) / Var(X).

Interpreting β₁: For each unit increase in X, Y is expected to increase by β₁ units, holding all else equal.

R²: The fraction of variance in Y explained by X. R² = 1 − (residual variance / total variance). R² = 0 means X explains nothing; R² = 1 means perfect fit.

Warning: R² is easy to inflate by adding more predictors. Adjusted R² penalizes for extra variables.

OLS Assumptions (and what breaks when they're violated)

1. Linearity: If the true relationship is non-linear, estimates are biased.
2. Independence: If errors are correlated (autocorrelation), standard errors are wrong.
3. Homoskedasticity: If variance of errors changes with X (heteroskedasticity), standard errors are wrong.
4. No perfect multicollinearity: If predictors are perfectly correlated, coefficients can't be uniquely estimated.
5. Normality of errors: Needed for small samples. Less critical for large samples (CLT).

For trading: Returns are not normally distributed (fat tails), and are autocorrelated in high-frequency contexts. This violates OLS assumptions. Solutions: robust standard errors, or switching to models that handle non-normality.

Overfitting in Practice

A model with 50 parameters fit to 100 data points will have artificially high R² — it has enough degrees of freedom to fit the noise. This is overfitting.

Signs of overfitting:

• High in-sample R² / Sharpe, low out-of-sample R²
• Coefficients that are huge or have unexpected signs
• Model performance degrades as you add more data

Prevention: regularization (Ridge, Lasso), cross-validation, fewer parameters relative to data.

4. Estimation and Confidence Intervals

Confidence Intervals

A 95% CI for the mean: x̄ ± 1.96 × (s/√n).

Common misinterpretation: "95% CI means there's a 95% chance the true mean is in this interval." Wrong — the true mean is either in the interval or not. The 95% refers to the long-run property: if you repeat the experiment many times, 95% of your intervals will contain the true mean.

For trading: A strategy with mean return 0.3% ± 0.5% (1σ) has a 95% CI of (−0.7%, 1.3%) for the daily return. This interval includes 0, meaning you can't reject the null of no edge.

Maximum Likelihood Estimation (MLE)

MLE finds the parameter values that make the observed data most probable.

For normally distributed data, the MLE of μ is the sample mean (x̄) and the MLE of σ² is the sample variance (with n in the denominator, not n−1 — which is why sample variance with n−1 is called the "unbiased" estimator, not the MLE).

Interview question: "You observe 10 coin flips: 7 heads. What's the MLE of p (probability of heads)?"

The likelihood of 7 heads in 10 flips as a function of p: L(p) = p⁷(1−p)³. Taking the derivative and setting to 0: p̂ = 7/10 = 0.7. The MLE is just the sample proportion.

5. Key Statistical Concepts for Trading

Sharpe Ratio

SR = (E[r] − r_f) / σ(r), where r_f is the risk-free rate.

Annualized SR = √252 × daily SR (using the square root of time scaling for independent daily returns).

What's a good Sharpe?

Sharpe	Assessment
< 0	Negative return
0–0.5	Weak; market-competitive but not compelling
0.5–1	Decent systematic strategy
1–2	Good strategy; most quant funds target this
2+	Excellent; possibly overfitted if backtested
3+	Extraordinary; very rare for real strategies

Detection problem: With daily Sharpe = 0.5, you need about (1.96/0.5)² × 252 ≈ 4 years of daily data to detect alpha at the 5% level with 50% power. With Sharpe = 1.0, it drops to about 1 year.

Autocorrelation

Correlation of a time series with a lagged version of itself. ACF(k) = Cor(Xₜ, Xₜ₋ₖ).

For returns: If returns are autocorrelated at lag 1 (ACF(1) > 0), yesterday's return predicts today's. This would be exploitable (momentum signal). Most equity returns at daily frequency have near-zero autocorrelation (efficient market hypothesis).

For mean reversion: Negative autocorrelation (ACF(1) < 0) implies mean reversion — after a big move, expect a partial reversal.

Stationarity

A time series is stationary if its statistical properties (mean, variance, autocorrelation) don't change over time.

Why it matters: Regression requires the predictor and response to be stationary. If you regress a non-stationary series on another non-stationary series, you can get spurious correlation.

Stock prices are NOT stationary (they have a trend and growing variance). Stock returns are approximately stationary (mean and variance roughly constant over time, ignoring regime changes).

The Central Limit Theorem

The sum of n iid random variables with mean μ and variance σ² converges to N(nμ, nσ²) as n → ∞.

Practical implication: Even if individual trade returns are not normally distributed (and they aren't — they have fat tails), the sum of many trades approaches a normal distribution. This is why portfolio-level risk is often modeled normally even when individual position risk is not.

6. Statistics Interview Questions

Q: What's the difference between frequentist and Bayesian statistics?

Frequentist: probability is the long-run frequency. Parameters are fixed (not random). Confidence intervals, p-values.

Bayesian: probability represents degrees of belief. Parameters have prior distributions; data updates them to posteriors. Credible intervals ("95% probability the parameter is in this range").

In trading: Bayesian is more natural (you're updating a belief about alpha), but frequentist tools are more standardized and auditable.

Q: What is the law of large numbers vs. the central limit theorem?

LLN: the sample mean converges to the population mean as n → ∞ (almost surely). About convergence.

CLT: the distribution of the sample mean is approximately normal for large n, regardless of the population distribution. About the shape of the distribution.

They're complementary: LLN says where the distribution is centered (at μ), CLT says the distribution's shape is normal.

Q: Why is maximum likelihood used instead of minimizing mean squared error?

They're equivalent for normally distributed errors. For non-normal distributions (e.g., fat-tailed returns), MLE adapts to the true distribution, while MSE implicitly assumes normality. MLE is more general.

Q: Your backtested strategy has a Sharpe of 3.0 over 6 months. Should you trade it?

No — this is almost certainly overfitting. A Sharpe of 3.0 is extraordinary. With only 6 months of data (126 daily returns), the standard error of the Sharpe estimate is approximately √(1 + SR²/2)/√n ≈ √(1 + 4.5)/√126 ≈ 0.66. A 95% CI for the true Sharpe is roughly (1.7, 4.3) — consistent with both excellent alpha and modest-but-inflated in-sample estimates.

The right answer: more out-of-sample data, and a detailed review of the research methodology for data leakage.

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Practice the probability and estimation categories daily to develop the statistical intuition that quant research interviews test. The daily drill reinforces calibration — the ability to match your confidence to your actual accuracy, which is exactly what statistical reasoning requires.