Topic 2.12 Notes – Sampling Distributions and the Central Limit Theorem

What a Sampling Distribution Is

A statistic is a number computed from sample data, like $\bar{x}$ or $\hat{p}$. Since different random samples give different data, the statistic can change too. That change is sampling variability.

A sampling distribution is the distribution of a statistic for all possible samples of one fixed size from one population using one sampling method.

That definition has a lot packed into it. Four things must stay fixed:

• the population
• the sample size
• the statistic
• the sampling method

What changes is the sample itself, so the statistic changes.

Three distributions to keep straight

The variable in a sampling distribution is the statistic itself, not the original observations.

Here is a quick example that puts all three ideas side by side.

• Population values might be $2, 4, 8$
• One sample might be $2, 8$
• The sampling distribution of $\bar{x}$ would contain possible sample means like $3, 5, 6$

In that last panel, the possible values are sample means from all samples of size 2, not individual data values.

You describe a sampling distribution the same way as any distribution: shape, center, spread, and unusual features. If you change the population, statistic, sample size, or sampling method, you change the sampling distribution.

Simulating a Sampling Distribution

Usually, listing all possible samples is impossible. So we simulate.

How the simulation works

1. Specify the population or probability model and any known parameter values.
2. Choose a fixed sample size $n$.
3. Generate one random sample using the correct process.
4. Calculate the statistic.
5. Record it.
6. Repeat many times.
7. Graph the recorded statistics and describe the distribution.

Each dot or bar on the graph represents one statistic from one simulated sample.

What must stay the same every time:

• same population model
• same sample size
• same statistic
• same sampling method, including with or without replacement

A common trap is mixing up sample size and number of repetitions.

• Sample size $n$ changes the sampling distribution itself.
• Repetitions only make your simulated graph smoother and more stable.

So if you simulate 5000 samples of size 10, you still have the sampling distribution for $n=10$, not for a larger sample.

When you describe a simulated sampling distribution, say:

• what statistic it shows
• what sample size was used
• what population/chance model was used
• the shape, center, and spread in context

Randomization Distributions

A randomization distribution comes from repeatedly reassigning observed response values to treatment groups.

This is a different idea from sampling distributions. You are not drawing new samples from a population.

What changes here

• In a simulated sampling distribution, you generate new random samples from a population model.
• In a randomization distribution, you keep the same observed responses and reshuffle the group labels.

How it is built

1. Keep the observed response values fixed.
2. Randomly reassign them to treatment groups, usually with the same group sizes.
3. Recalculate the statistic.
4. Record it.
5. Repeat and graph the results.

This picture shows one shuffle. The same response values are kept, but they get reassigned to new groups and produce a new difference in means.

One round of shuffling in a randomization test

The graph below the shuffle shows possible statistic values caused by random assignment alone.

Common AP confusion:

• calling the reassignment process “sampling”
• thinking each point is a new set of responses instead of a new grouping of the same responses

The Central Limit Theorem

The Central Limit Theorem says the sampling distribution of a sample mean $\bar{x}$ can be approximated by a normal distribution, and the approximation improves as $n$ gets larger.

This topic is about sample means, not every statistic.

• If the population is normal, the sampling distribution of $\bar{x}$ is normal for any sample size.
• If the population is not normal, larger $n$ makes the sampling distribution of $\bar{x}$ more bell-shaped and symmetric.
• Larger $n$ also makes sample means less spread out.

The picture below is the classic CLT pattern. It starts with a right-skewed population, then shows how the sampling distribution of $\bar{x}$ changes as $n$ increases.

Central Limit Theorem for sample means

Notice that the center stays near the population mean, but the sampling distributions become more symmetric and tighter as the sample size grows.

There is no universal cutoff. $n \ge 30$ is only a rough guideline.

• symmetric population with no outliers → moderate $n$ may be enough
• strongly skewed population → need larger $n$
• extreme outliers or heavy tails → may need much larger $n$

Randomness and independence still matter. A huge sample does not fix biased data collection.

What the CLT Does Not Mean

This is where students lose easy points.

• The population does not become normal.
• One large sample of raw data does not have to look normal.
• The CLT does not apply to every statistic.
• Larger samples reduce sampling variability, but they do not erase it.
• More simulation repetitions do not make $n$ larger.

Also keep CLT separate from the law of large numbers:

• Law of large numbers means a statistic tends to settle near the true value.
• CLT means the distribution of sample means across repeated samples tends toward normality.