Stats Coding in Interviews

import math

# ---- THE LIBRARY WAY (not what interviews want) ----
from scipy import stats
t_stat, p_value = stats.ttest_ind(group_a, group_b)
# One line. Zero understanding demonstrated.

# ---- THE FROM-SCRATCH WAY (what interviews want) ----
def ttest_ind_from_scratch(a, b):
    """Two-sample t-test, assuming unequal variances (Welch's t-test)."""
    n_a, n_b = len(a), len(b)
    mean_a = sum(a) / n_a
    mean_b = sum(b) / n_b

    # Sample variances (Bessel's correction: divide by n-1)
    var_a = sum((x - mean_a) ** 2 for x in a) / (n_a - 1)
    var_b = sum((x - mean_b) ** 2 for x in b) / (n_b - 1)

    # Standard error of the difference in means
    se = math.sqrt(var_a / n_a + var_b / n_b)

    # t-statistic
    t_stat = (mean_a - mean_b) / se

    # Welch-Satterthwaite degrees of freedom
    numerator = (var_a / n_a + var_b / n_b) ** 2
    denominator = (var_a / n_a) ** 2 / (n_a - 1) + (var_b / n_b) ** 2 / (n_b - 1)
    df = numerator / denominator

    return t_stat, df

# This shows you understand every component:
# - Why we divide by n-1 (Bessel's correction for sample variance)
# - What the standard error of a difference in means is
# - Why Welch's t-test uses adjusted degrees of freedom
# - The formula for Welch-Satterthwaite approximation

# Interviewer evaluation rubric for statistics coding:

evaluation_criteria = {
    "mathematical_correctness": {
        "weight": "CRITICAL",
        "description": "Does the implementation match the mathematical definition?",
        "red_flag": "Using n instead of n-1 for sample variance",
        "green_flag": "Correctly applying Bessel's correction and explaining why"
    },
    "from_scratch_ability": {
        "weight": "HIGH",
        "description": "Can the candidate implement without library calls?",
        "red_flag": "Reaching for scipy.stats immediately",
        "green_flag": "Writing the formula first, then coding each component"
    },
    "numerical_stability": {
        "weight": "HIGH",
        "description": "Does the candidate handle edge cases?",
        "red_flag": "Computing variance as E[X^2] - (E[X])^2 (catastrophic cancellation)",
        "green_flag": "Using Welford's algorithm or two-pass approach"
    },
    "edge_case_awareness": {
        "weight": "MEDIUM",
        "description": "Does the candidate consider degenerate inputs?",
        "red_flag": "Division by zero when n=1, empty lists",
        "green_flag": "Checking sample size, handling edge cases gracefully"
    },
    "complexity_analysis": {
        "weight": "MEDIUM",
        "description": "Can the candidate analyze time/space complexity?",
        "red_flag": "O(n^2) when O(n) is possible",
        "green_flag": "Streaming/online algorithms where applicable"
    }
}

# ---- FORMULAS YOU MUST KNOW ----

# Mean: sum of values divided by count
# mu = (1/n) * sum(x_i)

# Variance (population): average squared deviation from mean
# sigma^2 = (1/n) * sum((x_i - mu)^2)

# Variance (sample): Bessel's correction
# s^2 = (1/(n-1)) * sum((x_i - x_bar)^2)

# Standard deviation: square root of variance
# sigma = sqrt(variance)

# Covariance: how two variables move together
# cov(X,Y) = (1/(n-1)) * sum((x_i - x_bar)(y_i - y_bar))

# Correlation (Pearson): normalized covariance
# r = cov(X,Y) / (std(X) * std(Y))

# Normal PDF: f(x) = (1 / sqrt(2*pi*sigma^2)) * exp(-(x-mu)^2 / (2*sigma^2))

# Bayes' theorem: P(A|B) = P(B|A) * P(A) / P(B)

# t-statistic: t = (x_bar - mu_0) / (s / sqrt(n))

# Chi-squared statistic: chi2 = sum((observed - expected)^2 / expected)

Course Structure:

Lesson 2: Descriptive Statistics (6 problems)
  - Mean, median, mode from scratch
  - Variance, standard deviation
  - Percentiles, histogram binning
  - Correlation coefficient
  - Running/streaming statistics

Lesson 3: Implement Distributions (5 problems)
  - Normal distribution (PDF, CDF, sampling)
  - Binomial PMF and CDF
  - Poisson distribution
  - Exponential distribution
  - Sampling: inverse CDF, Box-Muller transform

Lesson 4: Hypothesis Testing in Code (5 problems)
  - Two-sample t-test (Welch's)
  - Chi-squared test of independence
  - A/B test significance analysis
  - Permutation test
  - Bootstrap confidence intervals

Lesson 5: Monte Carlo Simulation (5 problems)
  - Estimate pi via random sampling
  - Option pricing (Black-Scholes Monte Carlo)
  - Probability puzzles via simulation
  - Markov chain Monte Carlo (Metropolis-Hastings)

Lesson 6: Bayesian Methods in Code (5 problems)
  - Bayesian inference engine
  - Beta-Binomial conjugate model
  - Naive Bayes classifier from scratch
  - Bayesian A/B testing
  - Thompson sampling for bandits

Lesson 7: Patterns & Tips
  - Statistical coding patterns cheat sheet
  - Numerical precision guide
  - Interview FAQ