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Statistics interview questions for quant roles

Quant research interviews lean on statistics: variance rules, standard errors, correlation, regression, the normal distribution and portfolio risk. Every answer below shows how to get there.

32 sample questions with solutions. Edge has 241 statistics problems to practice, with a daily puzzle, timed games and a mock assessment. Practice them free.

1For a constant a, Var(aX) equals…

  1. a · Var X
  2. a² · Var X
  3. Var X
  4. |a| · Var X

Warm-up

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Answer: a² · Var X

Var(aX) = a² Var X.

2The correlation between two random variables always lies between…

  1. 0 and 1
  2. −1 and 1
  3. −∞ and ∞
  4. 0 and ∞

Warm-up

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Answer: −1 and 1

|Corr| ≤ 1 by the Cauchy–Schwarz inequality.

3The central limit theorem says the mean of many iid draws with finite variance is approximately…

  1. uniform
  2. normal
  3. exponential
  4. Poisson

Warm-up

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Answer: normal

Standardized sample means converge to a normal distribution.

4A Poisson(λ) random variable has variance…

  1. λ
  2. λ²
  3. √λ
  4. 1/λ

Warm-up

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Answer: λ

For a Poisson distribution, mean = variance = λ.

5Increasing a sample from 100 to 400 observations cuts the standard error of the mean by a factor of…

  1. 4
  2. 2
  3. 16
  4. √2

Warm-up

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Answer: 2

√(400/100) = 2.

6Bayes' rule: P(A | B) equals…

  1. P(B | A) P(A) / P(B)
  2. P(A) P(B)
  3. P(B | A)
  4. P(A) / P(B)

Warm-up

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Answer: P(B | A) P(A) / P(B)

P(A|B)P(B) = P(B|A)P(A).

7A Type I error is…

  1. rejecting a true null hypothesis
  2. failing to reject a false null
  3. a coding bug
  4. a biased sample

Warm-up

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Answer: rejecting a true null hypothesis

Type I is a false positive; Type II is a miss.

8The expected value of a Binomial(n, p) random variable is…

  1. np
  2. np(1 − p)
  3. n/p
  4. p/n

Warm-up

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Answer: np

Linearity: n × p.

9X is standard normal. What is the correlation between X and X²?

  1. 1
  2. 0
  3. −1
  4. 0.5

Warm-up

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Answer: 0

Cov = E[X³] = 0 by symmetry, even though they are dependent.

10To annualize a Sharpe ratio computed from daily returns, multiply by…

  1. 252
  2. √252
  3. 1/252
  4. 12

Warm-up

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Answer: √252

Sharpe scales with √T, so multiply by √252.

11For independent X and Y, Var(X + Y) equals…

  1. Var X + Var Y
  2. Var X − Var Y
  3. (Var X)(Var Y)
  4. 2 Cov(X,Y)

Warm-up

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Answer: Var X + Var Y

Var(X + Y) = Var X + Var Y + 2Cov, and Cov = 0.

12The standard error of a sample mean of n independent draws with SD σ is…

  1. σ
  2. σ/n
  3. σ/√n
  4. σ√n

Warm-up

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Answer: σ/√n

Var(mean) = σ²/n, so SE = σ/√n.

13X and Y are independent N(0,1). X + Y is distributed as…

  1. N(0,1)
  2. N(0,2)
  3. N(0,4)
  4. N(0,√2)

Warm-up

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Answer: N(0,2)

Normal with mean 0 and variance 2.

14A strategy looks great in backtest but fails live. The most likely cause is…

  1. too little data
  2. overfitting
  3. low volatility
  4. rounding

Warm-up

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Answer: overfitting

Fitting noise in-sample is overfitting; out-of-sample performance exposes it.

15An even-money bet wins 55% of the time. What fraction of your bankroll does the Kelly criterion say to bet?

Warm-up

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Answer: 1/10

f* = p − q = 0.55 − 0.45 = 0.10, so bet 10% of your bankroll each time.

16X and Y are independent with Var(X) = 4 and Var(Y) = 4. What is Var(1X + 1Y)?

VarianceWarm-up

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Answer: 8

Two rules: constants come out squared, and independent variances add with no covariance term. So 1² × 4 + 1² × 4 = 4 + 4 = 8.

17Draws have standard deviation 5. What is the standard error of the mean of 16 independent draws?

SamplingWarm-up

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Answer: 5/4

Averaging n independent draws divides the variance by n, so the spread shrinks by √n: 5/√16 = 5/4 = 1.25.

18Z is standard normal. What is P(Z > 0.5)? (Decimal.)

Normal distributionWarm-up

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Answer: 0.3085

Anchor on values worth memorizing: P(Z > 1) ≈ 0.159, P(Z > 1.645) ≈ 0.05, P(Z > 1.96) ≈ 0.025, P(Z > 2) ≈ 0.023, P(Z > 3) ≈ 0.0013. For z = 0.5: 1 − Φ(0.5) ≈ 0.3085.

19X ~ Binomial(10, 0.1). What is E[X]?

DistributionsWarm-up

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Answer: 1

X counts successes in 10 independent trials. Each trial contributes p = 0.1 on average, and expectations add: 10 × 0.1 = 1.

20X ~ Binomial(10, 0.1). What is Var(X)?

DistributionsWarm-up

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Answer: 9/10

Each trial is a 0/1 variable with variance p(1 − p) = 0.1 × 0.9 = 0.09. The trials are independent, so variances add: 10 × 0.09 = 0.9.

21Cov(X, Y) = 2, SD(X) = 2, SD(Y) = 4. What is Corr(X, Y)?

CorrelationWarm-up

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Answer: 1/4

Correlation is covariance with both spreads divided out: ρ = Cov/(SD(X) × SD(Y)) = 2/(2 × 4) = 0.25. It always lands between −1 and 1.

22X and Y are independent with Var(X) = 4 and Var(Y) = 4. What is Var(1X − 1Y)?

VarianceWarm-up

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Answer: 8

Two rules: constants come out squared, and independent variances add with no covariance term (so the minus sign disappears: (−1)² = 1). So 1² × 4 + 1² × 4 = 4 + 4 = 8.

23Draws have standard deviation 5. What is the standard error of the mean of 64 independent draws?

SamplingWarm-up

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Answer: 5/8

Averaging n independent draws divides the variance by n, so the spread shrinks by √n: 5/√64 = 5/8 = 0.625.

24What is the variance of a Uniform(0,1) random variable?

  1. 1/2
  2. 1/3
  3. 1/12
  4. 1/4

Interview

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Answer: 1/12

1/3 − 1/4 = 1/12.

25Adding another variable to an OLS regression never decreases…

  1. adjusted R²
  2. in-sample R²
  3. out-of-sample accuracy
  4. the p-values

Interview

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Answer: in-sample R²

In-sample R² can only rise or stay the same; adjusted R² can fall.

26Two assets have correlation −1. You can combine them into a portfolio with variance…

  1. zero
  2. their average
  3. their sum
  4. it depends on the mean

Interview

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Answer: zero

Weights inversely proportional to their volatilities give zero variance.

27A 95% confidence interval means…

  1. there's a 95% chance the true value is in this interval
  2. 95% of intervals built this way would contain the true value
  3. 95% of the data lie inside it
  4. the p-value is 0.95

Interview

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Answer: 95% of intervals built this way would contain the true value

It's a statement about the procedure's long-run coverage.

28Correlation stays the same under which change to X?

  1. Squaring it
  2. Multiplying by a positive constant and adding a constant
  3. Taking logs
  4. Multiplying by −1

Interview

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Answer: Multiplying by a positive constant and adding a constant

Positive affine maps leave correlation unchanged; −1 flips its sign.

29A portfolio holds 25% in asset A (vol 10%) and 75% in asset B (vol 15%), correlation 0.5. What is the portfolio volatility, as a decimal?

PortfoliosInterview

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Answer: 0.1269

Portfolio variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ = 0.000625 + 0.01266 + 0.002813 = 0.01609. Take the square root for volatility: ≈ 0.1269. Because ρ < 1, that's below the weighted average of the two vols (0.1375): diversification.

30X is normal with mean 100 and SD 10. What is P(X > 70)? (Decimal.)

Normal distributionInterview

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Answer: 0.9987

Standardize: z = (70 − 100)/10 = -3. Below the mean, so P(X > 70) = 1 − P(Z > 3) ≈ 0.9987.

31A strategy averages 2 basis points a day with daily volatility 0.5%. What is its annualized Sharpe ratio (252 trading days, zero risk-free rate)?

PerformanceInterview

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Answer: 0.635

Daily Sharpe = mean ÷ volatility = 0.02% / 0.5% = 0.04. Over a year the mean grows with the number of days (×252) but volatility only with its square root (×√252), so the ratio grows by √252 ≈ 15.87: 0.04 × 15.87 ≈ 0.635.

32Cov(X, Y) = 2 and SD(X) = 2. What is the OLS slope when regressing Y on X?

RegressionInterview

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Answer: 1/2

The OLS slope is Cov(X, Y)/Var(X) = 2/2² = 2/4 = 0.5. On average Y moves that much for each one-unit move in X.