Understanding Uncertainty · Mini-Lesson · pairs with Class 04 & Class 10

One Sample, One Point

Two distributions, not one — and exactly how the first collapses into the second

Understanding Uncertainty · in-class group work

A standalone mini-lesson in a small, growing set on estimator properties — and the one worth understanding before the bootstrap, since bootstrapping is this exact collapse, run thousands of times on resampled data instead of fresh die rolls. The running example is still the fair die, \(\theta = 3.5\).

A companion exploration built to accompany the sample-mean material in Class 04 and the sampling-distribution material in Class 10 — no external dataset, the widget generates its own die rolls in the browser.

New vocabulary is underlined like this — hover or tap any term for a one-line explanation that appears right below the line you are reading.

Think · Pair · Share · which distribution is which

Think · 2 minRoll a die 10 times and look at the 10 numbers you got — that's one distribution. Now imagine doing that whole 10-roll experiment 200 separate times and writing down just the average of each batch of 10 — that's 200 numbers, a second distribution. In what way are these two distributions about completely different things, even though both came from the same die?

I've done the Think step — reveal Pair & Share

Pair · 3 minCompare: what does a single value in the first distribution represent? What does a single value in the second represent?
Share · 2 minAgree on one sentence distinguishing them, then check it against the widget below.

The top histogram is one sample — the actual N die rolls you drew, plotted as a histogram of raw values 1 through 6. Averaging those N numbers collapses the whole histogram into a single number, \(\bar X_N\). Draw a new sample and that single number joins a second population of points, the sampling distribution of \(\bar X_N\), shown in the strip below against the true \(\theta = 3.5\). Click any dot in that strip to bring its original N-event histogram back up top.

Top: the histogram of one sample's N raw rolls. Bottom: every sample's mean collapsed to one dot each, plus the solid θ̂ line — the average of every dot drawn so far. The dashed connector threads the currently selected dot back to the histogram that produced it.

Same die, two different distributions, two different questions. The top histogram answers "what did this one batch of rolls look like?" — a question about N individual events. The bottom strip answers "how much does the average of a batch like this move around, from batch to batch?" — a question about T collapsed summaries. Every dot in the bottom strip erases the top histogram's internal detail down to a single number; clicking a dot is the only way to get that detail back.

N and T are two different "more data" knobs — don't let them blur together. Growing N (a bigger sample) tightens each individual dot, \(\bar X_N\), around θ — that's the spread of the sampling distribution shrinking. Growing T (more samples drawn) tightens θ̂, the running average of every dot so far, around θ instead — a completely separate estimate, built by averaging averages. Use "Draw 10 new samples" a few times and watch θ̂'s solid line settle down even while N, and each individual dot's spread, stay exactly the same.

Check your intuition

Draw five or six new samples at N = 10, then drag N up to 100 and draw a few more. What visibly changes about the bottom strip — and what stays exactly the same about the top histogram's basic shape?

Show answer

The bottom strip's dots pull in tighter around θ = 3.5 as N grows — that's \(\bar X_N\) becoming more precise. The top histogram's shape doesn't fundamentally change — it's still six bars near a rough 1/6-each split, just built from more rolls. Growing N makes the average more precise; it doesn't make any single die roll less random.

FAQ

How is this different from the bootstrap?

It's the same collapse, with one change: here, every new sample is N fresh die rolls from the real population. In the bootstrap, you only have one real sample, so instead you resample from that one sample, with replacement, and treat the resample's mean the same way — one more dot in the same kind of bottom strip. If this page's "collapse" step doesn't feel automatic yet, the bootstrap will be much harder to trust.

Why does clicking an old dot still show the right histogram? Doesn't redrawing overwrite the data?

Each sample's N raw rolls are stored when it's drawn, not thrown away — only the display collapses them to a dot. "Collapsing" here is something you choose to do to summarize a sample, not something that destroys the underlying data.

More estimator-properties mini-lessons: Unbiased, but never consistent What the sample mean minimizes Why the sample variance undershoots How bad is my estimator? One sample, one point (skewed) One sample, one point (variance)