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Video status: Related supplementary video included; the original lecture recording has not been verified. Source details
This is article 17 in Reading Stanford CS109, covering Summer 2026 Lecture 16: Bootstrapping on July 20 with Chris Gregg. It follows the current worksheet, answer key, LLM Learning Guide, and reader chapters on bootstrapping and samples. Current slides are unavailable and Canvas video is gated, so this remains an L2 artifact guide.
The worksheet, key, and guide are all three pages. The formal agenda is P1–P6 plus challenge, with no orphan page or numbering gap. A sample distribution describes data spread; a sampling distribution describes how a statistic changes when the whole experiment is repeated. Error bars target the latter.
P1: Review a CLT for checkout totals
One hundred IID checkout counts each have mean three and variance four. For total S,
S ≈ N(100×3,100×4)=N(300,400), SD=20The counts are integer-valued, so S≥320 begins at continuous boundary 319.5:
P(S≥320)≈1-Φ((319.5-300)/20)
=1-Φ(0.975)≈0.165P2: Ping Times mean, unbiased variance, and SE
The eight latencies [12,15,11,14,13,19,12,16] sum to 112, giving X̄=14 ms. Squared deviations from 14 sum to 48:
S² = 48/(8-1) = 48/7 ≈ 6.86 ms²
SE(X̄)=√(S²/8)≈0.93 msReplacing the unknown population mean with the fitted sample mean systematically shrinks squared deviations; Bessel's n-1 correction makes the variance estimator unbiased. Sample SD describes individual-ping spread, while SE estimates the standard deviation of the mean across repeated eight-ping experiments.
The approximate CLT interval is
14±2(0.93)=[12.1,15.9] msAcross repeated sampling, this construction covers the true mean roughly 95% of the time. It does not assign 95% probability to a fixed parameter lying in this already computed interval.
P3: A tiny bootstrap that can be enumerated
For sample [2,4,9], three with-replacement draws create 3³=27 ordered resamples. The probability the maximum equals nine is
P(max=9)=1-(2/3)³=19/27≈0.70The resample mean equals original mean five exactly when its sum is 15. Only multiset {2,4,9} works, with 3!=6 orderings:
P(mean=5)=6/27=2/9≈0.22With replace=False, drawing n from n only permutes the original sample. Every statistic is identical, collapsing the bootstrap distribution to one point. Replacement preserves the empirical PMF's draw probabilities.
P4: Bootstrap a median without a closed-form SE
For 50 app ratings,
medians=[]
repeat 10,000 times:
resample=choice(ratings,50,replace=True)
medians.append(median(resample))
return std(medians)The bootstrap assumes the observed sample histogram is a useful proxy for the population distribution. Resampling it then mimics fresh experiments and works for medians, variances, IQRs, and differences of means.
It becomes unreliable when data are non-IID because naive resampling destroys dependence, or when a long-tailed population contains consequential rare extremes absent from the original sample.
P5: A null bootstrap test for a compiler flag
Flag A has 40 runtimes with mean 210 ms; B has 45 with mean 204 ms, an observed six-ms gap. The null says both groups come from one runtime distribution and the gap is sampling noise.
Pool all 85 measurements under the null, then redraw original group sizes:
universal=concat(A_times,B_times)
count=0
repeat 10,000 times:
A*=choice(universal,40,replace=True)
B*=choice(universal,45,replace=True)
if abs(mean(A*)-mean(B*))>=6: count+=1
p=count/10,000p=0.03 means that, if the null were true, about 3% of experiments like this would produce a gap at least this extreme. It is not P(null=true|data) and does not measure effect size. With p=0.4, the gap is ordinary under the null and does not establish a real flag difference.
P6: Bootstrap uncertainty in a Course Size estimator
Ten historical ratios ri are IID, and final enrollment is T=300R. Plug-in estimates are
Ê[T] = 300R̄ = 300×(1/10)Σri
Var-hat(T)=300²S_R²
=300²×(1/9)Σ(ri-R̄)²The target in part (c) is not Var-hat(T) but its sampling variability with only ten historical observations:
estimates=[]
repeat 10,000 times:
r*=choice([r1,...,r10],10,replace=True)
estimates.append(300²×sample_variance(r*))
return std(estimates)Each round must recompute the full target statistic. Bootstrapping only raw-ratio SD would answer a different question.
Challenge: Two schools of statistical thought
Fifteen flips yield five heads and ten tails. Starting from Laplace prior Beta(2,2), the Bayesian route gives
p|data~Beta(7,12), E[p|data]=7/19≈0.37The frequentist bootstrap treats five ones and ten zeros as the empirical sample, repeatedly draws 15 values with replacement, and records head fraction. Ten thousand fractions approximate the sampling distribution of p-hat, centered near observed 5/15=1/3.
The distributions answer different questions. A Bayesian posterior expresses belief about random parameter p given prior and data. A bootstrap distribution treats p as a fixed unknown and describes estimator p-hat across repeated samples.
How to use the LLM Learning Guide
The six concepts are population/sample/statistic, unbiased variance, SE, the bootstrap algorithm, arbitrary statistics and failure modes, and null p-values. First state which quantity varies across repeated experiments, then write resampling code. A hypothesis test additionally requires samples from the null-implied pool and a conditional interpretation of the resulting p-value.
Material boundaries
- This guide covers P1–P6, the optional challenge, and all six guide concepts; all three pages and problem numbers are complete.
- Current slides are unavailable and video is gated; an L2 article does not reconstruct missing lecture content.
- Bootstrap code is equivalent organization of official pseudocode, not a claim about production implementation details.
- The limited artifact scope qualifies for the short-material exception; the article remains
draft: truepending independent review.
Course video sources
Verified the official Summer 2026 archive: the course is recorded and its syllabus says recordings are available through Canvas. The archived homepage's Videos link redirects to Stanford Canvas login and requires course access, so there is no public Summer 2026 player. The 2022 Stanford Online public recording(s) below cover the same topic but come from a different offering with different lecture numbering; they are supplementary material, not Summer 2026 lecture recordings.
Original videos: Stanford CS109 Probability for Computer Scientists I Bootstraping and P-Values I 2022 I Lecture 19
Official sources:
- CS109 Summer 2026 — Canvas lecture recordings (login required)
- CS109 Summer 2026 — archived official schedule
- CS109 Summer 2026 — archived syllabus and recording policy
- Stanford Online CS109 2022 public playlist (different offering, supplementary)
Checked: 2026-10-10.
Content check: verified against the video transcript (2026-10-10): Read the transcript of 2022 Lecture 19, "Bootstrapping and P-Values" (about 70K characters): it starts with sample variance and standard error, then bootstrapping, and introduces the null hypothesis and p-values near the end, matching this lecture. The transcript has no median or compiler-flag examples, so P4 and P5 in this post are not video content.
Update Log
- 2026-10-10: Added explicit video status and checked recording sources and access notes.
- 2026-10-10: Rechecked video status. Switched to related supplementary video: embedded the 2022 Stanford Online public recording on the same topic; the Summer 2026 original is behind Canvas sign-in and could not be verified.
- 2026-10-10: Checked the video content against its transcript. The video topic matches; the median and compiler-flag problems are not video content, as the check note says.
References
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