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Week 10 · Chapter

Chapter 10 — Sampling Distributions & the Central Limit Theorem

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
The module's primary reading · ~30–40 minutes · Objective 5 · Read this before (or alongside) the week's lecture — everything else in the module builds on it.

Why you already care about this

Pick up a snack-size candy bag. The label says 200 grams. Weigh it and you'll get something like 197.2. Weigh the next one: 203.1. A third: 199.4. Nobody's bag is exactly 200 — and yet the company prints the number with a straight face, regulators accept it, and you've never once felt cheated enough to check. Somehow a promise about an average survives the fact that every individual bag breaks it.

The same quiet miracle is everywhere. A parcel service promises an average delivery time while individual packages bounce all over. A call center reports its average hold time. An elevator placard — "capacity 9 people or 1,800 lb" — is a bet about the average weight of whoever steps in. Individuals are wildly variable; averages, mysteriously, are steady enough to build promises on.

This chapter explains the miracle, and it is the hinge of the entire course. By the end you will be able to: describe sampling variability and say precisely what a sampling distribution is; give the center and spread of the sample mean's distribution — including the single most important formula of the second half, the standard error σ/√n — and keep SD vs. SE straight; state the Central Limit Theorem and say when it applies; compute the probability that a sample mean (or a sample proportion) lands in any region; and explain, in plain language, why averages deserve more trust than individual values — and exactly how much more.

Here is the one-line version to carry all week: one value is a guess; an average is a promise with a wobble you can compute.

Where we are

Weeks 1–8 built the whole toolkit: honest data collection (Week 1), description (Weeks 2–3), relationships (Week 4), probability (Weeks 5–7), and the normal model with its z-scores and table (Week 8). The midterm closed that half. This week the course turns around and points the toolkit at its real target: what a sample can tell us about the population it came from. Week 8's bell curve is about to show up somewhere you have no right to expect it — and next week it becomes the confidence interval.

1. Sampling variability, and the idea that tames it

Draw a random sample and compute its mean: you get a number. Draw a second sample, same honest method: you get a different number. You have seen this with your own hands — in the Week 1 data lab, no two students' random penguin samples gave the same mean, and drawing a fresh sample changed your own answer. That bounce is called sampling variability, and the first thing to internalize is that it is not a mistake. It is what sampling does.

The week's big mental move is to stop staring at one sample and ask a bigger question: what is the complete collection of answers my sampling method could give? Imagine — you never actually do this; you imagine it — taking every possible random sample of size n from the population and computing x̄ for each one. Pile all those x̄'s into a single distribution. That pile is the sampling distribution of the sample mean: the full catalog of every value x̄ could take, and how often. It is the distribution your one real x̄ was drawn from.

Notice what kind of object this is. Back in Week 6 you met random variables — quantities whose values are determined by chance, with distributions, expected values, and SDs. Before you draw your sample, x̄ is a random variable, and the sampling distribution is its distribution.

Three distributions are now in play, and keeping them straight is half the week's work:

  1. The population distribution — every individual value in the world. Any shape at all.
  2. The distribution of one sample — your n measured values. Its histogram resembles the population (bumps, skew, and all; bigger n makes the resemblance sharper, not more bell-shaped).
  3. The sampling distribution of x̄ — one entry per sample, not per individual. A new distribution with its own center, spread, and shape — the subject of this chapter.

Worked example — the four-bag toy build. Take a tiny population you can hold in one hand: four candy bags weighing 196, 198, 202, and 204 g. The population mean is μ = (196 + 198 + 202 + 204) ÷ 4 = 800 ÷ 4 = 200 g. Now list every possible sample of n = 2 bags (order doesn't matter, so there are six) and compute each sample's mean:

  • {196, 198} → x̄ = 197
  • {196, 202} → x̄ = 199
  • {196, 204} → x̄ = 200
  • {198, 202} → x̄ = 200
  • {198, 204} → x̄ = 201
  • {202, 204} → x̄ = 203

The pile {197, 199, 200, 200, 201, 203} is the sampling distribution of x̄ for this little world, written out completely. Two features deserve a long look:

  • Its average is exactly μ. (197 + 199 + 200 + 200 + 201 + 203) ÷ 6 = 1200 ÷ 6 = 200. Sample means don't systematically run high or low; misses above and misses below balance perfectly. We say x̄ is an unbiased estimator of μ.
  • It huddles. The individual bags span 196 to 204 — a range of 8 g. The sample means span only 197 to 203 — a range of 6 g. Averaging pulls extremes inward, because a lucky-high bag tends to be paired with an ordinary one.

Any one sample mean can miss the truth. But the collection of possible sample means is centered exactly on the truth and huddles around it. The next section says how tightly — with a formula.

Misconception check. "So the sample mean equals the population mean?" No — unbiased means the sampling distribution is centered on μ, with no systematic lean. Your particular x̄ almost certainly misses μ. What we're about to get is a ruler for the typical size of that miss.

2. The center and spread of x̄: meet the standard error

For a random sample of size n from any population with mean μ and standard deviation σ, the sampling distribution of x̄ has:

  • Center: mean = μ. (Unbiased — the toy build showed you why.)
  • Spread: standard deviation = σ ⁄ √n — a quantity so important it carries its own name: the standard error (SE) of the sample mean.

Set off, because the whole second half of the course stands on it:

The two rulers. σ is the ruler for individuals — how far ONE value typically strays from μ. σ ⁄ √n is the ruler for averages — how far a sample MEAN of n values typically strays from μ. Same units, completely different jobs. Before any calculation, ask: am I being asked about one value, or about an average?

Worked example — the candy machine. A filling machine drops candy into snack bags whose individual weights average μ = 200 g with σ = 8 g, in an approximately bell-shaped way (filling machines are like that). The bags ship in cases of n = 25, and the quality team tracks each case's average bag weight.

  • SE of the case average: σ ⁄ √n = 8 ⁄ √25 = 8 ⁄ 5 = 1.6 g.
  • In words: one bag typically sits about 8 g from 200. A case's average typically sits only about 1.6 g from 200.

That is the label's secret. The company cannot control your individual bag — an 8-gram wobble is baked into the machine. But it doesn't promise your bag; it promises the average, and averages of 25 barely move. Individuals wobble; averages behave.

The √n in the denominator deserves respect, because it runs every study budget in the world. With n = 25, the SE is 8/5 = 1.6. Want half that wobble — an SE of 0.8? You need √n = 10, which means n = 100. Quadruple the sample to halve the noise. Doubling n only divides the wobble by √2 ≈ 1.41. Precision gets expensive fast — which is why real polls stop near 1,500 people and real studies agonize over sample size.

Misconception check. "SD, SE — aren't those basically the same?" This is the most consequential mix-up of the second half, so catch it now. σ describes individuals; σ/√n describes averages of n. Using σ where the SE belongs — the wrong-ruler error — makes averages look far more variable than they are, and it will quietly wreck probability answers, confidence intervals, and tests if you let it. The cure is a habit: write the ruler on its own line ("ruler = 8/√25 = 1.6") before any z-score.

One honesty note. The four-bag build sampled without replacement from a tiny population, so the ÷√n law doesn't apply to it exactly — from the toy we read only "centered at μ" and "narrower." The σ/√n formula is exact for independent draws and works beautifully whenever the population dwarfs the sample, which is our standing situation (and the reason real pollsters don't fret about sampling one person "out" of a nation).

In practice (your tools). Nothing new to buy: =AVERAGE() and =STDEV() computed both rulers' ingredients back in Week 3. What's new is the divide: in a spreadsheet, the SE of a planned sample mean is just =8/SQRT(25). This week's data lab has you earn that formula — you'll draw 30 real samples, bank 30 sample means, and watch =STDEV() of your means land right where σ/√n predicted.

3. The Central Limit Theorem

We know the sampling distribution's center (μ) and spread (σ/√n). One question remains — its shape — and the answer is the most famous theorem in statistics.

The Central Limit Theorem (CLT). Take random samples of size n from any population with mean μ and standard deviation σ. As n grows, the sampling distribution of x̄ becomes approximately normal — with mean μ and standard error σ/√n — no matter what shape the population has.

Read that again, because it has no business being true. The population can be skewed, lumpy, two-humped, or plain weird. The individual data never stop looking that way. But pile up sample means and a bell emerges — out of nowhere, from pure arithmetic. Week 8 handed you a curve that turns distances into percentages; the CLT says that curve governs averages universally, which is why one table and one z-recipe will power every inference tool from here to Week 18.

The theorem comes with fine print — three lines' worth:

  • Population already normal? Then x̄ is exactly normal at every sample size, even n = 2.
  • Population any other shape? The working rule of thumb is n ≥ 30 for the bell to be trustworthy — and the heavier the skew or the wilder the outliers, the more you want. (Nothing is magic about 30; the magic word is approximately.)
  • Always required: a genuinely random sample. The CLT repairs shape, never bias. A convenience sample's mean follows a lovely bell — centered on the wrong value. Week 1's slogan survives intact: method beats size.

You have already met the evidence. In the Week 8 lab, the pooled penguin body masses were not one bell — the histogram was right-skewed with a second bump, three species mixed together. The CLT's audacious claim is that if you draw repeated samples of those very penguins and histogram the sample means, the bell appears anyway. This week's data lab has you build exactly that, by hand, thirty samples deep — the single best afternoon of "seeing is believing" in the course.

Misconception check. "The CLT says my data will look normal if I collect enough." No — and this one fools students and chatbots. Your one sample's histogram mirrors the population: skewed population, skewed sample, at n = 100 or n = 10,000. The population never changes — it doesn't know you're sampling it. What goes normal is the distribution of sample means across samples. Say the theorem's subject out loud every time: the sampling distribution of the sample mean.

4. Using the bell: probabilities for sample means

With shape (normal), center (μ), and spread (σ/√n) in hand, every "how likely is this average?" question becomes a Week 8 problem with a new ruler:

z = (x̄ − μ) ⁄ (σ ⁄ √n) — "sample mean, minus the center, divided by the standard error."

The friendly z-table rides again (same values as Week 8 — left-tail areas, verified):

The table below gives the left-tail area (proportion below) for each z value used this week.

z Area below z Area below
−2.5 0.0062 0.5 0.6915
−2.0 0.0228 1.0 0.8413
−1.5 0.0668 1.25 0.8944
−1.25 0.1056 1.5 0.9332
−1.0 0.1587 2.0 0.9772
−0.5 0.3085 2.5 0.9938

(z = 0 sits at 0.5000, and the empirical rule's 68–95–99.7 still applies — to whatever is normal, which now includes sample means.)

Worked example — one threshold, two rulers. The candy machine again: individual bags approximately normal, μ = 200 g, σ = 8 g; cases of n = 25; SE = 1.6 g. How often is a weight over 204 g? Ask it both ways and watch the rulers separate:

  • One random bag over 204 g. The subject is an individual, so the ruler is σ = 8: z = (204 − 200) ⁄ 8 = 0.5 → area above = 1 − 0.6915 = 0.3085 ≈ 31%. Happens all the time.
  • A case's average over 204 g. The subject is a mean of 25, so the ruler is the SE, 1.6: z = (204 − 200) ⁄ 1.6 = 2.5 → area above = 1 − 0.9938 = 0.0062 = 0.62%. Genuinely rare.

Same threshold — 31% versus 0.62%, a 50-fold drop. Averages don't merely wobble less; extreme averages become rare. Hold onto that: every inference tool ahead works by noticing that some observed average would be rare if a claim were true.

Worked example — the elevator problem. A building's service elevator placard reads: capacity 9 people or 1,800 lb. Nine adults step in. Suppose adults in this building have weights approximately normal with μ = 190 lb and σ = 24 lb, and treat the riders (for now) as a random sample.

  1. Turn the total into a mean. The total exceeds 1,800 lb exactly when the average rider exceeds 1,800 ⁄ 9 = 200 lb. One divide converts a sum question into a mean question — no new machinery needed.
  2. Find the ruler. SE = 24 ⁄ √9 = 24 ⁄ 3 = 8 lb.
  3. Standardize and read. z = (200 − 190) ⁄ 8 = 1.25 → area above = 1 − 0.8944 = 0.1056 ≈ 10.6% — about one fully-loaded ride in nine.

That is not a comfortable number, and it is exactly why real placards sit far below what the cables can actually hold: engineers run this calculation on averages and then stack safety margins on top.

One professional caveat, stated plainly: real elevator riders are not a random sample — people board in groups (a visiting team, a family), so weights arrive correlated, and the true risk can sit above or below 10.6%. The formula is only as good as the random-sampling assumption feeding it. Checking conditions before trusting output is the habit that separates statistics from arithmetic — you'll do it in every remaining week.

In practice (your tools). For P(case average < 204): =NORM.DIST(204, 200, 1.6, TRUE) returns 0.9938 — and the third argument must be the SE (1.6), never σ (8). That argument box is where the wrong-ruler error hides in software; the function has no idea whether you're asking about a bag or a case, so you must feed it the right ruler. Desmos-class tools: define normaldist(200, 1.6) and shade — the SD slot takes the SE for any sample-mean question.

5. The sample proportion p̂ rides the same math

Much of the data you care about is yes/no: on time or late, satisfied or not, clicked or didn't. The sample proportion p̂ summarizes it — and here's the trick that makes this section short: a proportion is a mean in disguise. Code each individual as 1 (success) or 0 (not), and p̂ is exactly the average of the 0s and 1s. So everything above transfers:

  • Center: the sampling distribution of p̂ has mean p — unbiased again.
  • Spread: SE = √( p(1 − p) ⁄ n ).
  • Shape: approximately normal when the sample expects at least 10 of each kind — np ≥ 10 and n(1 − p) ≥ 10 (the success/failure condition; you watched this bell assemble itself from binomial bars in Week 7).

Worked example — the on-time rate. A parcel service's long-run on-time rate is p = 0.80. Each day, quality control pulls a random sample of n = 100 tracking scans and computes the day's p̂.

  • Center: 0.80. SE = √(0.80 × 0.20 ⁄ 100) = √0.0016 = 0.04.
  • Shape check: np = 80 ✓ and n(1 − p) = 20 ✓ — the bell applies.
  • How often would a day show 75% or worse purely by sampling luck? z = (0.75 − 0.80) ⁄ 0.04 = −1.25 → area below = 0.1056 ≈ 10.6%.

About one day in nine looks that bad with nothing wrong at all. A manager who launches an investigation at every 75% day is chasing sampling noise; a manager who knows the SE knows which dips deserve a second look. (Week 13 turns exactly this instinct into a formal hypothesis test.)

Misconception check. "SE of p̂ = p(1 − p)/n." That's the variance — the SE wears the square root. Sanity-check with units of wobble: for p = 0.80 and n = 100, an "SE" of 0.0016 would mean day-to-day proportions glued to 0.800 ± 0.003ish, which contradicts the 75% days you just computed. When a wobble looks impossibly small, hunt for a missing √ — it's Week 3's σ-vs-σ² error in new clothes.

Key terms

  • Sampling variability — the sample-to-sample bounce in a statistic's value; a feature of sampling, not an error.
  • Sampling distribution — the distribution of a statistic's values across all possible random samples of size n.
  • Unbiased — a statistic whose sampling distribution is centered exactly on the parameter it estimates (x̄ for μ; p̂ for p).
  • Standard error (SE) — the standard deviation of a statistic's sampling distribution; for x̄ it is σ/√n, for p̂ it is √(p(1 − p)/n).
  • SD vs. SE — σ is the ruler for individuals; σ/√n is the ruler for sample means.
  • Central Limit Theorem (CLT) — for random samples from any population, the sampling distribution of x̄ becomes approximately normal as n grows, with mean μ and SE σ/√n.
  • n ≥ 30 rule of thumb — the working sample size at which the CLT's bell is trustworthy for non-normal populations (more under heavy skew; any n if the population is normal).
  • Sampling distribution of p̂ — approximately normal with mean p and SE √(p(1 − p)/n) when np ≥ 10 and n(1 − p) ≥ 10.
  • Success/failure condition — the np ≥ 10 and n(1 − p) ≥ 10 check for using the normal shape with p̂.

Summary

  • Honest samples disagree — that's sampling variability. The fix is to study the whole catalog of possible answers: the sampling distribution.
  • The sampling distribution of x̄ is centered exactly at μ (unbiased) with spread σ/√n — the standard error. Two rulers: σ for individuals, σ/√n for averages. Quadruple the sample to halve the noise.
  • The Central Limit Theorem: sample means go approximately normal as n grows, whatever the population's shape — n ≥ 30 for arbitrary shapes, any n if the population is normal, and always a random sample. The CLT fixes shape, never bias.
  • Probabilities about a sample mean: z = (x̄ − μ)/(σ/√n), then the Week 8 table. Totals convert to means by dividing by n. Extreme averages are rare — the lever all of inference pulls.
  • A proportion is a mean of 0s and 1s: p̂ is approximately normal with mean p and SE √(p(1 − p)/n) once np ≥ 10 and n(1 − p) ≥ 10.

Check your understanding

Work these before looking at the answers — they're ungraded, and the answers (with reasoning) are at the bottom of the page.

  1. A machine fills fruit-chew pouches with mean 150 g and SD 15 g, approximately normally. Pouches ship in boxes of 25, and the packer tracks each box's average pouch weight. (a) Give the center and standard error of the sampling distribution of the box average. (b) Using the chapter's z-table, find the probability a box's average is below 146.25 g.
  2. Three histograms are described. Histogram A: wide and right-skewed, built from 500,000 values. Histogram B: wide and right-skewed-ish, built from the 200 values in one random sample. Histogram C: narrow and bell-shaped, built from the means of many random samples of 200. Match A, B, and C to population distribution, distribution of one sample, and sampling distribution of x̄ — and say what gives C away.
  3. A warehouse cargo lift is rated for 16 crates or 4,000 lb. Crate weights at this warehouse are approximately normal with mean 240 lb and SD 40 lb, and the 16 crates on a pallet can be treated as a random sample. What is the probability the lift's rated load is exceeded? (Convert the total to a mean first.)
  4. A courier app's records show that 64% of all recipients leave delivery instructions (p = 0.64). A dispatcher pulls a random sample of 144 of today's orders. (a) Verify the sampling distribution of p̂ is approximately normal. (b) Find the probability that 72% or more of the sampled orders have instructions.

Answers (worked reasoning — read after attempting)

  1. (a) Center = 150 g (the sampling distribution of x̄ is centered at μ — unbiased). SE = σ/√n = 15 ⁄ √25 = 15 ⁄ 5 = 3 g. (b) The subject is a box average, so the ruler is the SE: z = (146.25 − 150) ⁄ 3 = −3.75 ⁄ 3 = −1.25 → area below = 0.1056 ≈ 10.6%. About one box in nine averages below 146.25 g by ordinary sampling wobble.
  2. A = the population distribution (every individual value; any shape; here right-skewed). B = the distribution of one sample (200 values that mirror the population's skew — one entry per individual). C = the sampling distribution of x̄ (one entry per sample). C is given away by two fingerprints: it is narrower (spread σ/√200 instead of σ) and bell-shaped even though the population is skewed — exactly what the CLT promises for means, and what no histogram of raw values would do.
  3. Totals-to-means first: the load is exceeded when the average crate tops 4,000 ⁄ 16 = 250 lb. Ruler: SE = 40 ⁄ √16 = 40 ⁄ 4 = 10 lb. Standardize: z = (250 − 240) ⁄ 10 = 1.0 → area above = 1 − 0.8413 = 0.1587 ≈ 16%. Roughly one fully-loaded pallet in six — a number a safety engineer would act on (and a reminder of why rated loads carry margins).
  4. (a) np = 144 × 0.64 = 92.16 ≥ 10 ✓ and n(1 − p) = 144 × 0.36 = 51.84 ≥ 10 ✓ — the normal shape applies, with center 0.64 and SE = √(0.64 × 0.36 ⁄ 144) = √0.0016 = 0.48 ⁄ 12 = 0.04. (b) z = (0.72 − 0.64) ⁄ 0.04 = 2.0 → area above = 1 − 0.9772 = 0.0228 ≈ 2.3%. A 72%-or-more day happens only about twice in a hundred samples by luck — rare enough that a dispatcher who sees it should suspect something real changed.

Next up in this module: the slide deck if you want the lecture's visual walk-through, then the Readings & Resources page — and when you're ready to practice, Lecture Tutorial 10 turns your chatbot into your personal sampling-distributions tutor.