Sampling and Estimation

Bootstrap Mean (Small)

Estimate the sampling distribution of the mean by averaging B given resamples. Each resample is a fixed literal list (values drawn with replacement from the original), making results deterministic without relying on RNG agreement between naive and library. Nested loops: outer over B=3 resamples, inner computes each resample mean; a final loop averages the resample means. Library: np.mean per resample then np.mean of those means, on identical data.

By hand

Original: [2, 4, 6] (mean=4.0). Three given resamples (n=3 each, drawn with replacement from the original): [2,2,4] → 2.667, [4,6,6] → 5.333, [6,4,2] → 4.0. Bootstrap mean = (2.667+5.333+4.0)/3 = 4.0. The nested loop makes the mean-per-resample step explicit.

naive.py
Replay: real traced execution (multi-file project)
original = [2, 4, 6]
resamples = [[2, 2, 4], [4, 6, 6], [6, 4, 2]]
B = len(resamples)
means = []
for rs in resamples:
    total = 0.0
    for v in rs:
        total = total + v
    means.append(total / len(rs))
boot_mean = 0.0
for m in means:
    boot_mean = boot_mean + m
boot_mean = boot_mean / B
print('RESULT:', round(boot_mean, 4))

  1. original ← [2, 4, 6]

    1original = [2, 4, 6]2resamples = [[2, 2, 4], [4, 6, 6], [6, 4, 2]]
    values this step[2, 4, 6]original

  2. resamples ← [[2, 2, 4], [4, 6, 6], [6, 4, 2]]

    1original = [2, 4, 6]2resamples = [[2, 2, 4], [4, 6, 6], [6, 4, 2]]3B = len(resamples)
    values this step[[2, 2, 4], [4, 6, 6], [6, 4, 2]]resamples

  3. B ← 3

    2resamples = [[2, 2, 4], [4, 6, 6], [6, 4, 2]]3B = len(resamples)4means = []
    values this step3B

  4. means ← []

    3B = len(resamples)4means = []5for rs in resamples:
    values this step[]means

  5. rs ← [2, 2, 4], total ← 8.0, v ← 4, means ← [2.6666666666666665]

    pass 1 of 3
    4means = []5for rs in resamples:6    total = 0.07    for v in rs:8        total = total + v9    means.append(total / len(rs))10boot_mean = 0.0
    values this step[2, 2, 4]rs4.0 → 8.0total2 → 4v[] → [2.6666666666666665]means
    All 3 passes — pass 1 is the card above
    passrstotalvmeans
    1[2, 2, 4]4.0 → 8.02 → 4[] → [2.6666666666666665]
    2[2, 2, 4] → [4, 6, 6]10.0 → 16.04 → 6[2.6666666666666665] → [2.6666666666666665, 5.333333333333333]
    3[4, 6, 6] → [6, 4, 2]10.0 → 12.04 → 2[2.6666666666666665, 5.333333333333333] → [2.6666666666666665, 5.333333333333333, 4.0]

  6. for rs in resamples:

    4means = []5for rs in resamples:6    total = 0.0

  7. boot_mean ← 0.0

    9    means.append(total / len(rs))10boot_mean = 0.011for m in means:
    values this step0.0boot_mean

  8. m ← 2.6666666666666665, boot_mean ← 2.6666666666666665

    pass 1 of 3
    10boot_mean = 0.011for m in means:12    boot_mean = boot_mean + m13boot_mean = boot_mean / B
    values this step2.6666666666666665m0.0 → 2.6666666666666665boot_mean
    All 3 passes — pass 1 is the card above
    passmboot_mean
    12.66666666666666650.0 → 2.6666666666666665
    22.6666666666666665 → 5.3333333333333332.6666666666666665 → 8.0
    35.333333333333333 → 4.08.0 → 12.0

  9. for m in means:

    10boot_mean = 0.011for m in means:12    boot_mean = boot_mean + m

  10. boot_mean ← 4.0

    12    boot_mean = boot_mean + m13boot_mean = boot_mean / B14print('RESULT:', round(boot_mean, 4))
    values this step12.0 → 4.0boot_mean

  11. stdout ← RESULT: 4.0

    13boot_mean = boot_mean / B14print('RESULT:', round(boot_mean, 4))
    values this stepRESULT: 4.0stdout

With the library

np.mean is applied per resample via a list comprehension, then np.mean averages those means. Same resamples as naive; the resample means are shown for comparison.

library.py 
import numpy as np
from dalib.display import set_display
set_display()

original = [2, 4, 6]
resamples = [[2, 2, 4], [4, 6, 6], [6, 4, 2]]
B = len(resamples)
resample_means = [float(np.mean(rs)) for rs in resamples]
boot_mean = float(np.mean(resample_means))
print('B:', B)
print('resample means:', [round(m, 4) for m in resample_means])
print('RESULT:', round(boot_mean, 4))

B: 3
resample means: [2.6667, 5.3333, 4.0]
RESULT: 4.0

Honesty

This lesson shows the computation exactly, on a tiny pinned sample. The arithmetic is correct and reproducible, but with a sample this small the result is not a valid statistical finding — it demonstrates the mechanism, not evidence. Real inference needs an adequate sample size and assumption checks (e.g. B≥1000 resamples and an adequate original n); the bootstrap estimate here should be read as "how the resampling is computed," not as a conclusion about a population.

Implementation notes

  • RNG parity rule: resamples are given as fixed literals rather than generated at runtime. Python's random and numpy's RNG produce different sequences even with the same integer seed, so live draws would break parity.
  • In practice B is typically 1000–10000. Here B=3 is enough to show the structure without exceeding the 60-event trace budget.
  • Bootstrap does not require knowing the population distribution. The resample variability estimates the sampling variability directly from the data.
  • Cross-reference: seeded-sample-mean (this chapter) for the mean computation each resample iteration applies; standard-error (this chapter) for the analytic SE that bootstrap approximates empirically.