Lecture: Sampling Distributions & the Central Limit Theorem
Lecture script · about 15–20 minutes spoken · Week 10
Opening
Imagine a hospital quality team looking at this month's average wait in the emergency room. It's a little higher than last month's. Is something wrong, or is that the ordinary wobble you'd get from a different set of patients? A polling team faces the same question when it surveys about 1,000 voters and wants to speak for millions. So does a field biologist who weighs a handful of birds and wants to say something about the whole colony. Each of them is asking one thing: how much would this average, or this share, shift if I drew a fresh sample?
That's today's question, and by the end of this lecture you'll have a real answer to it.
Here's where we are. In Week 1 you learned the difference between a population and a sample, and between a parameter and a statistic. In Week 3 you met the mean and the standard deviation. In Weeks 5 through 7 you built probability, random variables, and the binomial. Week 8 gave you z-scores, and last week, in Week 9, you met the normal model and the empirical rule. Today we put those pieces together. A sample mean is itself a random variable, and its distribution turns out to look like that normal model. Next week, in Week 11, this becomes a confidence interval for a mean.
We'll go in eight steps: why samples disagree, a sampling distribution you can build by hand, its center and spread, the standard error, the Central Limit Theorem, using it, proportions, and finally a real population of penguins.
1. Sampling variability: why a second sample gives a different answer
Let's start with a fair die. Roll it 30 times and average the faces. Then roll it 30 more times and average again. The two averages won't match. Nothing went wrong. Each set of rolls caught a different mix of high and low faces.
Here's the everyday version. A food truck owner pulls 5 receipts at random one morning and gets an average of 11 dollars. The next morning, 5 more random receipts average 13 dollars. The truck didn't change overnight. The samples did.
That wobble has a name. Sampling variability is the natural change in a statistic, such as a sample mean or a sample proportion, from one random sample to the next, even when every sample comes from the same population by the same method.
Keep the two roles from Week 1 in mind. The die's true mean is 3.5. That's a parameter, the population mean, written mu. It never moves. The average of your 30 rolls is a statistic, the sample mean, which we write as x̄ and read "x bar". It moves every time you roll again. For the die you also know the population standard deviation, called sigma. It's about 1.71 points, and it tells you how far a single roll typically lands from 3.5.
Pause and try this. You roll a fair die 30 times twice and get two different averages. Which number stayed fixed the whole time?
Think about what belongs to the die itself.
Here's the answer. The population mean, 3.5, stayed fixed. The two averages are statistics, and statistics wobble from sample to sample.
2. A sampling distribution you can build by hand: averages of two rolls
To study the wobble, you need a list of every value the statistic can take, with its chance. A sampling distribution is the distribution of the values a statistic takes across all possible random samples of the same size from the same population; it shows how much the statistic varies from sample to sample and where it centers.
For most real populations, you can't list every possible sample. But for the smallest case, you already have the list. Back in Week 5 you built the table of sums for two fair dice. An average of two rolls is that sum divided by 2.
So the average of two rolls can be 1, 1.5, 2, and so on up to 6, 11 possible values in all. An average of 1 needs a sum of 2, which happens 1 time in 36. An average of 3.5 needs a sum of 7, which happens 6 times in 36, about 16.7% of the time. The chances climb from 1 out of 36 up to 6 out of 36, then fall back down.
Picture two histograms side by side. One roll gives six bars of equal height, a flat shape. Two-roll averages give a triangle, tallest in the middle at 3.5 and shortest at the ends, 1 and 6. Averaging only two values already pulls the extremes toward the middle.
Pause and try this. Which is more likely for an average of two rolls: exactly 3.5, or exactly 1?
Look back at the chances for each.
Here's the answer. Exactly 3.5. It has 6 chances in 36, while an average of 1 has only 1 chance in 36.
3. Center and spread: the sampling distribution has mean mu and standard deviation sigma over root n
Two rules describe the sampling distribution of x̄. First, its center is the population mean, mu. Averages don't aim too high or too low on purpose. Second, its standard deviation is sigma divided by the square root of n, where n is the sample size, the number of values in one sample.
Let's test both rules on the triangle. The sum of two dice has expected value 7, so the average has mean 3.5. Same as mu. Good. The sum has a standard deviation of 2.4152, so the average has half of that, about 1.21 points. Now the rule: sigma over the square root of 2 is also about 1.21 points. Both routes agree.
Now let each sample be 30 rolls. The center is still 3.5. The spread is sigma, 1.7078, divided by the square root of 30, which is about 5.48. That gives about 0.3118 points, or roughly 0.31. So a single roll typically misses 3.5 by about 1.71 points, but a 30-roll average misses by only about 0.31 points. That's about 5.5 times tighter.
Here's a picture that helps. Think of a crowd of people guessing the number of jelly beans in a jar. Single guesses are all over the place. But if you average groups of guesses, the highs and lows cancel each other out, and the group averages bunch up tight around the middle.
Pause and try this. Which wobbles more around 3.5: one roll of a die, or the average of 30 rolls?
Compare 1.71 with 0.31.
Here's the answer. One roll wobbles more. Averaging cancels highs against lows, so the 30-roll average stays much closer to 3.5.
4. Standard deviation versus standard error: two different wobbles
We've now used two different spreads, and they answer two different questions. Sigma answers, "How much do single values vary?" Sigma over the square root of n answers, "How much does the sample mean vary from sample to sample?" The second one gets its own name. The standard error is the standard deviation of a statistic's sampling distribution: how much the statistic typically wobbles from sample to sample. For a sample mean it is sigma over the square root of n, and it shrinks as the sample size n grows. In real data you rarely know sigma, so you'll use the sample standard deviation, s, in its place. That version takes center stage next week.
Let's use last week's commute model. Suppose single trip times follow a normal model with a mean of 32 minutes and a standard deviation of 6 minutes. One trip varies by about 6 minutes. Now average 4 trips. The standard error is 6 over the square root of 4, which is 6 over 2, or 3 minutes. Average 36 trips, and it's 6 over 6, only 1 minute.
Notice the pattern. Going from 1 trip to 4 trips cut the wobble in half, from 6 minutes to 3. To halve the standard error, you need four times the sample. Precision gets expensive.
Pause and try this. For the commute model, which number describes one trip, and which describes the average of 4 trips: 6 minutes or 3 minutes?
Ask yourself: one value, or an average?
Here's the answer. 6 minutes describes one trip, and 3 minutes, the standard error, describes the average of 4 trips.
5. The Central Limit Theorem: averages pile into a bell
Go back to the die. One roll is flat. Two-roll averages make a triangle. Keep going to 30-roll averages, and the triangle rounds off into a smooth, symmetric bell centered at 3.5. That isn't a quirk of dice. It's one of the most useful results in all of statistics.
The Central Limit Theorem is the result that, for random samples of size n from a population with mean mu and standard deviation sigma, the sampling distribution of x̄ is approximately normal when n is large, whatever the population's shape. It is the distribution of the mean, not the data, that becomes approximately normal.
So how large is large? The course uses a working rule. The CLT rule of thumb says n is large enough at about 30 for a moderately skewed population, fewer for a roughly symmetric one, and any n at all when the population is assumed to follow a normal model. It's a rule of thumb, not a law of nature.
Here's the trap of the week. Suppose you collect 100 purchases from a bookstore where most purchases are small and a few are large. The histogram of those 100 purchases still leans right, like the population. Your data didn't become a bell. What becomes a bell is the pattern of averages across many samples. Here's one more caution. The theorem needs random samples, and no sample size fixes a biased method.
Pause and try this. A classmate says, "I have 50 values, so my data are normal now." What should the sentence say instead?
Think about what the theorem is really about.
Here's the answer. It should say the sampling distribution of x̄ is approximately normal. The 50 data values keep the shape of the population they came from.
6. Use it: judge whether one sample mean is surprising
Now the payoff. Once you know the center, the standard error, and the shape, last week's tools work on averages. The z-score uses the standard error where sigma used to go.
Take 30-roll averages of a fair die. The center is 3.5 and the standard error is about 0.31. Two standard errors below 3.5 is 2.88, and two above is 4.12. By the empirical rule, about 95% of 30-roll averages land in that range. So an average outside roughly 2.9 to 4.1 would be surprising for a fair die. If a friend's 30 rolls averaged 4.6, you'd have good reason to wonder about the die.
Now the commute. A single trip of 38 minutes has a z-score of 1, since it's 6 minutes above 32, and the standard deviation is 6. The standard normal table says 0.8413 of the area sits below a z of 1, so about 15.9% of single trips take longer than 38 minutes. Now take the average of 4 trips. The standard error is 3 minutes, so an average of 38 is 6 over 3, a z-score of 2. By the empirical rule, only about 2.5% of four-trip averages run that long. That's about 6 times rarer than one long trip.
Pause and try this. For a four-trip average, should the z-score divide by 6 minutes or by 3 minutes?
Remember which spread belongs to an average.
Here's the answer. By 3 minutes, the standard error. Dividing by 6 treats the average like one trip and makes a rare average look ordinary.
7. Proportions: the sampling distribution of p̂
Lots of questions are about a share, not an average. What share of voters favor a proposal? The same ideas carry over. The population proportion is the share of a whole population that has a trait of interest, written p. The sample proportion, written p̂ and read "p hat", is the share of a sample that has the trait. Its sampling distribution is centered at p, its standard deviation is the square root of p times 1 minus p, over n, and it's approximately normal when the expected successes and expected failures are both at least 10. That's the same large-counts rule of thumb you met in Week 7.
Suppose 50% of a large population favors a proposal, so p is 0.5. Poll 1,000 people at random. The standard deviation of p̂ works out to about 0.0158. Two of those is about 0.03, so about 95% of such polls land within about 3 percentage points of the true 50%. The check passes easily: 500 expected successes and 500 expected failures.
Here's the surprising part. The size of the population doesn't appear anywhere in that formula. That's why a well-drawn poll of about 1,000 people can speak for millions. The wobble depends on the size of the sample, not the size of the country. And the same rule from before holds: a poll of 100 has a spread of 0.05, while a poll of 2,500 has a spread of 0.01. That's 25 times the people for one-fifth of the wobble.
Pause and try this. 12 rolls of a die have about 2 expected sixes. Can you use a normal model for the proportion of sixes?
Compare the expected count with 10.
Here's the answer. No. With only 2 expected sixes, below 10, the distribution is lumpy and piled against zero.
8. A real population to sample from: the penguins
Dice and polls are tidy. Real data are messier, so this week's Data Lab tries the same ideas on the penguins you've followed all term. You'll treat the 342 measured body masses as if they were the whole population. Their mean is 4,201.75 grams. Treated as a population, their standard deviation is about 800.78 grams, a hair below the sample version you met earlier, 801.95 grams.
Now imagine drawing a random sample of 30 penguins and averaging their masses. The standard error is about 800.78 over the square root of 30, or about 146.20 grams. So single penguins differ from the mean by about 800 grams, while averages of 30 typically differ by about 146 grams. The population is mildly right-skewed, so by the rule of thumb, means of samples of 30 should pile up in a roughly normal shape centered near 4,201.75 grams. In the lab you'll check that prediction with your own spreadsheet.
One honest note. Treating 342 penguins as a population is a teaching move. In the real field study, they're a sample from much larger colonies.
Pause and try this. In the Data Lab, will the 30 masses in one sample look like a bell?
Think about which shape the theorem promises.
Here's the answer. No. One sample looks like the population, mildly right-skewed. The bell shows up only in the averages of many samples.
Closing
Let's pull it together in the order we built it. A statistic wobbles from sample to sample, while the parameter stands still. A sampling distribution lists every value the statistic can take, and for two dice rolls it's a triangle. Its center is mu, and its spread is sigma over the square root of n. That spread is the standard error, which is different from the standard deviation of single values, and four times the sample halves it. The Central Limit Theorem says averages of large samples follow an approximately normal model, whatever the population's shape, though the data themselves keep their shape. That lets you judge whether one average is surprising. Sample proportions behave the same way once the large-counts check passes. And the penguins give you a real population to test it all on.
Next in this module come the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, and the quiz, then the discussion, the assignment, and the Data Lab. Next week, in Week 11, Confidence Intervals for a Mean, the standard error becomes the margin around an estimate, and you'll build your first confidence interval.
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