Probability and Statistics · Sampling

• Population: everyone you care about. Sample: the part you actually measure. A parameter describes the population (μ), and a statistic describes the sample (x̄).
• Simple random sampling: everyone has an equal chance of being picked.
• Stratified sampling: sample within groups (strata) so each group is represented. It reduces variance when the groups differ.
• Bias: a biased method (convenience samples, non-response) stays biased no matter how large the sample is. More data doesn’t fix a bad design.

The bootstrap resamples the sample with replacement many times and recomputes the statistic each time. The spread of those values estimates the standard error, with no formula needed. scipy.stats.bootstrap does this for you.

import numpy as np
from scipy import stats

rng = np.random.default_rng(4)
# population: two branches with different mean scores
pop_a = rng.normal(70, 8, 8000)     # 80% of students
pop_b = rng.normal(85, 5, 2000)     # 20% of students
population = np.concatenate([pop_a, pop_b])
print("true mean", round(population.mean(), 2))

srs = rng.choice(population, 200, replace=False)
strat = np.concatenate([rng.choice(pop_a, 160, replace=False), rng.choice(pop_b, 40, replace=False)])
biased = rng.choice(pop_b, 200, replace=False)          # only surveyed one branch
print("random", round(srs.mean(), 2), "| stratified", round(strat.mean(), 2), "| biased", round(biased.mean(), 2))

res = stats.bootstrap((srs,), np.mean, n_resamples=5000, random_state=rng)
lo, hi = res.confidence_interval
print("bootstrap SE", round(res.standard_error, 3), "| 95% CI", round(lo, 2), "to", round(hi, 2))

Output:

true mean 73.13
random 73.12 | stratified 73.06 | biased 84.82
bootstrap SE 0.71 | 95% CI 71.77 to 74.54