The Central Idea

A sample statistic is not fixed before the sample is drawn.

If we repeatedly take samples of the same size from the same population, the observations will differ and the statistic will usually differ as well.

The sampling distribution of a statistic is the probability distribution of that statistic over all possible samples generated by a specified sampling process.

Sampling distributions tell us how much a statistic tends to vary and where it tends to be centered.

From One Sample to Repeated Sampling

Suppose we draw a random sample of size \(n\) and compute a sample mean \(\bar{x}\).

We obtain one number.

If we return to the population and repeat the entire sampling process many times, we obtain

\[ \bar{x}_1,\quad \bar{x}_2,\quad \bar{x}_3,\quad \ldots \]

The distribution of those possible sample means is the sampling distribution of the sample mean.

A Statistic Is a Random Variable

Before a random sample is selected, the value of a statistic is unknown.

Because its value depends on random selection, the statistic can be treated as a random variable.

For example,

\[ \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i \]

denotes the sample mean as a random variable before the sample is observed.

After the data are collected, a realized numerical value is often written as \(\bar{x}\).

Population Distribution, Sample Distribution, and Sampling Distribution

Three different distributions must be distinguished.

These distributions answer different questions.

Example: Three Different Distributions

Suppose the variable is weekly study time among all students at a university.

The population distribution describes weekly study time for every student.

One sample distribution describes weekly study time for the students in a particular random sample.

The sampling distribution of \(\bar{X}\) describes the sample means that would arise across all possible random samples of the specified size.

The Center of a Sampling Distribution

A central question is whether a statistic tends to be centered at the population quantity it is intended to estimate.

For a simple random sample from a population with mean \(\mu\),

\[ \boxed{E(\bar{X})=\mu}. \]

The sample mean is therefore an unbiased estimator of the population mean under the standard sampling model.

Unbiasedness

An estimator \(\hat{\theta}\) of a population parameter \(\theta\) is unbiased when

\[ \boxed{E(\hat{\theta})=\theta}. \]

Unbiasedness is a property of the estimator's sampling distribution.

It does not mean that every individual sample estimate equals the true parameter.

It means that the estimator is correctly centered across repeated sampling.

Sampling Variability

Even an unbiased statistic generally changes from one sample to another.

The spread of its sampling distribution measures this sampling variability.

Statistics with less sampling variability tend to provide more precise estimates of the population parameter.

Standard Error

The standard deviation of a sampling distribution is called the standard error of the statistic.

In general,

\[ \boxed{\operatorname{SE}(\hat{\theta}) =\sqrt{\operatorname{Var}(\hat{\theta})}}. \]

Standard error describes how much the estimator typically changes from sample to sample under the sampling process.

Standard Deviation versus Standard Error

A population or sample standard deviation describes variability among individual observations.

A standard error describes variability among values of a statistic across repeated samples.

These are related ideas, but they refer to different distributions.

Sampling Distribution of the Sample Mean

Suppose \(X_1,\ldots,X_n\) are independent observations from a population with mean \(\mu\) and finite variance \(\sigma^2\).

The sample mean is

\[ \bar{X}=\frac{X_1+\cdots+X_n}{n}. \]

Its expected value is

\[ E(\bar{X})=\mu. \]

Its variance is

\[ \operatorname{Var}(\bar{X})=\frac{\sigma^2}{n}. \]

Standard Error of the Sample Mean

Taking the square root of the variance gives

\[ \boxed{\operatorname{SE}(\bar{X})=\frac{\sigma}{\sqrt{n}}}. \]

This formula reveals one of the most important scaling relationships in statistics.

The typical sample-to-sample fluctuation of the mean decreases in proportion to \(1/\sqrt{n}\).

Why the Mean Becomes More Stable

Averaging combines information from multiple observations.

Independent positive and negative deviations from the population mean tend to offset one another.

As \(n\) grows, the average therefore tends to fluctuate less than an individual observation.

The formula

\[ \frac{\sigma}{\sqrt{n}} \]

quantifies that stabilization.

Example: Standard Error of a Mean

Suppose a population has standard deviation

\[ \sigma=18. \]

For samples of size

\[ n=36, \]

the standard error of the sample mean is

\[ \operatorname{SE}(\bar{X}) =\frac{18}{\sqrt{36}} =\frac{18}{6} =3. \]

Sample means typically vary on a scale much smaller than individual observations.

Effect of Increasing Sample Size

If the sample size is multiplied by \(4\), the standard error of the mean is divided by \(2\):

\[ \frac{\sigma}{\sqrt{4n}} =\frac{\sigma}{2\sqrt{n}}. \]

If the sample size is multiplied by \(9\), the standard error is divided by \(3\).

Precision therefore improves with sample size, but according to a square-root relationship rather than a linear one.

Exact Normal Sampling Distribution

If the population itself is normal,

\[ X\sim N(\mu,\sigma^2), \]

then the sample mean has an exactly normal sampling distribution:

\[ \boxed{ \bar{X}\sim N\left(\mu,\frac{\sigma^2}{n}\right) }. \]

This result holds for every positive sample size \(n\) under independent sampling.

The Central Limit Theorem

Many populations are not normally distributed.

Even so, under broad conditions, the standardized sample mean approaches a normal distribution as the sample size grows.

The central limit theorem states that

\[ \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \]

approaches a standard normal distribution as \(n\) becomes large, assuming the observations satisfy the required independence and finite-variance conditions.

What the Central Limit Theorem Does Not Say

The central limit theorem does not say that the original population becomes normal.

It concerns the distribution of a properly standardized statistic across repeated samples.

The population can remain skewed, discrete, multimodal, or otherwise nonnormal.

How Large Must the Sample Be?

There is no universal sample size that guarantees an excellent normal approximation for every population.

The needed size depends on population shape.

Rules such as \(n\ge 30\) are rough classroom heuristics, not universal mathematical guarantees.

Standardizing the Sample Mean

When the normal approximation is appropriate and \(\sigma\) is known, we can standardize the sample mean:

\[ \boxed{ Z=\frac{\bar{X}-\mu}{\sigma/\sqrt{n}} }. \]

This measures how many standard errors the observed sample mean lies above or below the population mean.

Example: Probability for a Sample Mean

Suppose a population has

\[ \mu=50,\qquad \sigma=12, \]

and samples of size

\[ n=36 \]

are taken.

Then

\[ \operatorname{SE}(\bar{X}) =\frac{12}{6} =2. \]

A sample mean of \(54\) is

\[ z=\frac{54-50}{2}=2 \]

standard errors above the population mean.

Sampling Distribution of a Sample Proportion

Suppose each sampled unit is classified as either a success or a failure.

Let the population success probability be \(p\), and let \(X\) be the number of successes in a sample of size \(n\).

The sample proportion is

\[ \hat{p}=\frac{X}{n}. \]

Under independent Bernoulli sampling,

\[ X\sim\operatorname{Binomial}(n,p). \]

Center of the Sample Proportion

The expected value of the sample proportion is

\[ \boxed{E(\hat{p})=p}. \]

Thus \(\hat{p}\) is an unbiased estimator of the population proportion under the standard Bernoulli sampling model.

Standard Error of the Sample Proportion

Under independent Bernoulli sampling,

\[ \operatorname{Var}(\hat{p}) =\frac{p(1-p)}{n}. \]

Therefore,

\[ \boxed{ \operatorname{SE}(\hat{p}) =\sqrt{\frac{p(1-p)}{n}} }. \]

Example: Standard Error of a Proportion

Suppose

\[ p=0.40 \]

and

\[ n=600. \]

Then

\[ \operatorname{SE}(\hat{p}) =\sqrt{\frac{0.40(0.60)}{600}} =\sqrt{0.0004} =0.02. \]

The sample proportion typically fluctuates on a scale of about \(0.02\), or two percentage points.

Normal Approximation for a Sample Proportion

Because the sample proportion is a scaled binomial count, its sampling distribution becomes approximately normal when the expected numbers of successes and failures are sufficiently large.

A common practical check is

\[ np\ge 10 \]

and

\[ n(1-p)\ge 10. \]

These thresholds are conventions for approximation quality, not exact mathematical boundaries.

Standardizing a Sample Proportion

When the normal approximation is appropriate,

\[ \boxed{ Z= \frac{\hat{p}-p} {\sqrt{p(1-p)/n}} }. \]

This expresses the observed sample proportion in standard-error units relative to the population proportion.

Example: Sample Proportion in Standard-Error Units

Suppose

\[ p=0.50,\qquad n=400. \]

Then

\[ \operatorname{SE}(\hat{p}) =\sqrt{\frac{0.5(0.5)}{400}} =0.025. \]

If a sample produces

\[ \hat{p}=0.55, \]

then

\[ z=\frac{0.55-0.50}{0.025}=2. \]

The observed proportion is two standard errors above \(p\).

Finite Population Sampling

The formulas

\[ \frac{\sigma}{\sqrt{n}} \]

and

\[ \sqrt{\frac{p(1-p)}{n}} \]

correspond naturally to independent sampling or to sampling from a population so large that removing sampled units has negligible effect.

When a substantial fraction of a finite population is sampled without replacement, a finite population correction can reduce the standard error.

Finite Population Correction

For a simple random sample without replacement from a finite population of size \(N\), a common correction factor is

\[ \boxed{ \sqrt{\frac{N-n}{N-1}} }. \]

For the sample mean, the corrected standard error is

\[ \boxed{ \operatorname{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}} }. \]

Example: Finite Population Correction

Suppose

\[ N=1000,\qquad n=200,\qquad \sigma=15. \]

Without the correction,

\[ \frac{15}{\sqrt{200}}\approx 1.061. \]

The finite population correction is

\[ \sqrt{\frac{1000-200}{999}} = \sqrt{\frac{800}{999}} \approx 0.895. \]

The corrected standard error is therefore approximately

\[ 1.061(0.895)\approx 0.950. \]

When the Finite Population Correction Is Small

If the sampling fraction

\[ f=\frac{n}{N} \]

is small, then

\[ \sqrt{\frac{N-n}{N-1}} \]

is close to \(1\).

In that situation, sampling without replacement behaves approximately like independent sampling for standard-error calculations.

Sampling Distribution of a Difference in Means

Suppose two independent samples estimate means from two populations.

The statistic

\[ \bar{X}_1-\bar{X}_2 \]

estimates

\[ \mu_1-\mu_2. \]

Its expected value is

\[ E(\bar{X}_1-\bar{X}_2) = \mu_1-\mu_2. \]

Standard Error of a Difference in Means

For independent samples with population standard deviations \(\sigma_1\) and \(\sigma_2\),

\[ \boxed{ \operatorname{SE}(\bar{X}_1-\bar{X}_2) = \sqrt{ \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2} } }. \]

Independent sources of variance add at the variance level.

Sampling Distribution of a Difference in Proportions

For two independent samples,

\[ \hat{p}_1-\hat{p}_2 \]

estimates

\[ p_1-p_2. \]

Its expected value is

\[ E(\hat{p}_1-\hat{p}_2) = p_1-p_2. \]

Standard Error of a Difference in Proportions

Under independent sampling,

\[ \boxed{ \operatorname{SE}(\hat{p}_1-\hat{p}_2) = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} } }. \]

Later lessons will distinguish the versions used for estimation from those used under specific null hypotheses.

Estimated Standard Errors

Population quantities such as \(\sigma\) and \(p\) are often unknown.

We therefore estimate the standard error using sample information.

For a sample mean,

\[ \widehat{\operatorname{SE}}(\bar{X}) = \frac{s}{\sqrt{n}}, \]

where \(s\) is the sample standard deviation.

For a sample proportion,

\[ \widehat{\operatorname{SE}}(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}. \]

Standard Error Is Not the Same as Estimation Error

The actual error of an estimate is the realized difference between the statistic and the unknown parameter.

For a mean,

\[ \bar{x}-\mu. \]

The standard error is not that realized difference.

Instead, it describes the typical scale of variation across repeated samples.

Shape, Center, and Spread

A useful way to analyze any sampling distribution is to ask three questions.

  1. What is its center?
  2. What is its spread?
  3. What is its shape?

The center is connected to bias.

The spread is connected to precision.

The shape determines which probability approximations and inferential tools are appropriate.

Bias and Variability Are Different

An estimator can be nearly unbiased but highly variable.

Another estimator can have low variability but be systematically shifted away from the true parameter.

These are different properties of a sampling distribution.

The next lesson will examine the interaction among bias, variability, and representativeness.

Precision

An estimator is often described as more precise when its sampling distribution has less spread.

For two unbiased estimators of the same parameter, the estimator with the smaller variance is generally more precise.

Precision therefore concerns repeatability across hypothetical samples, not whether one observed estimate happens to be close to the truth.

Consistency

An estimator is consistent when it approaches the true parameter as sample size increases, under the assumptions of the model.

Informally,

\[ \hat{\theta}_n \to \theta \]

as

\[ n\to\infty. \]

The sample mean is a central example under standard conditions.

Simulation of a Sampling Distribution

Sampling distributions can often be explored computationally.

A simulation workflow is:

  1. specify a population or probability model;
  2. draw a random sample of size \(n\);
  3. compute the statistic;
  4. record the value;
  5. repeat the process many times;
  6. graph and summarize the recorded statistics.

Simulation Reveals the Distribution of the Statistic

Suppose we repeat the sampling process \(10{,}000\) times.

If we calculate a sample mean each time, the resulting \(10{,}000\) means approximate the sampling distribution of \(\bar{X}\).

The histogram of those means can reveal:

Exact Distribution versus Simulated Distribution

An exact sampling distribution is derived from the full probability model.

A simulated sampling distribution approximates that distribution by repeated computational sampling.

Simulation is especially useful when the exact distribution is difficult to derive or when students want to visualize the repeated-sampling idea.

Example: Simulating Sample Means

Suppose a population is strongly right-skewed.

We repeatedly take samples of size \(n=5\) and plot their means.

The resulting sampling distribution may still be noticeably skewed.

If we repeat the simulation with \(n=50\), the distribution of sample means will often appear much more nearly normal and much narrower.

This illustrates both the central limit effect and the decrease in standard error.

Sampling Distributions and Confidence Intervals

Confidence intervals use the sampling distribution of an estimator to quantify plausible uncertainty around an observed estimate.

A generic confidence-interval structure is

\[ \boxed{ \text{estimate} \pm \text{critical value}\times \text{standard error} }. \]

The details depend on the estimator, sampling model, and approximation used.

Sampling Distributions and Hypothesis Tests

Hypothesis testing asks how unusual an observed statistic would be if a specified null model were true.

That question requires a reference distribution for the statistic under the null hypothesis.

In many classical procedures, that reference distribution is a sampling distribution or an approximation to one.

Sampling Distributions and P-Values

A p-value measures how extreme the observed result is relative to a reference distribution generated under a null hypothesis.

Without a sampling distribution, the phrase “unusually far from the null value” has no quantitative meaning.

Later lessons will formalize this logic.

Sampling Distributions Depend on the Design

The same statistic can have different sampling behavior under different sampling designs.

Simple random sampling, stratified sampling, cluster sampling, unequal probability sampling, and dependent observations can produce different variances.

Standard-error formulas must therefore match the actual design rather than being applied mechanically.

Sampling Distributions Depend on the Statistic

Different statistics have different sampling distributions.

The sample mean, median, variance, proportion, correlation coefficient, regression slope, maximum, and quantiles do not generally share the same distribution.

Statistical theory studies each statistic according to its own probability behavior.

Sampling Distribution of the Sample Variance

For a normal population, the sample variance has a particularly important exact relationship.

If

\[ X_1,\ldots,X_n \]

are independent normal observations with variance \(\sigma^2\), then

\[ \boxed{ \frac{(n-1)S^2}{\sigma^2} \sim \chi^2_{n-1} }. \]

This connects variance estimation to the chi-square distribution.

Student's t Distribution

When the population is normal but \(\sigma\) is unknown, replacing \(\sigma\) with the sample standard deviation \(S\) changes the reference distribution.

The statistic

\[ \boxed{ T= \frac{\bar{X}-\mu}{S/\sqrt{n}} } \]

follows a Student's \(t\) distribution with

\[ n-1 \]

degrees of freedom.

Why the t Distribution Has Heavier Tails

Replacing the unknown population standard deviation with the random sample standard deviation introduces additional uncertainty.

The \(t\) distribution accounts for that uncertainty with heavier tails than the standard normal distribution.

As the degrees of freedom increase, the \(t\) distribution approaches the standard normal distribution.

Degrees of Freedom

Degrees of freedom measure, in a model-dependent way, how many independent pieces of information remain after certain constraints are imposed.

For the usual one-sample variance estimate,

\[ S^2 = \frac{1}{n-1} \sum_{i=1}^{n}(X_i-\bar{X})^2, \]

the deviations from the sample mean satisfy

\[ \sum_{i=1}^{n}(X_i-\bar{X})=0. \]

Once \(n-1\) deviations are known, the last is determined.

Bootstrap Sampling Distributions

When an exact sampling distribution is unavailable or inconvenient, resampling methods can approximate uncertainty from the observed data.

In the bootstrap, we repeatedly sample with replacement from the observed sample and recompute the statistic.

The resulting bootstrap distribution approximates features of the statistic's sampling distribution under suitable conditions.

Bootstrap Workflow

  1. Begin with an observed sample of size \(n\).
  2. Draw \(n\) observations with replacement from that sample.
  3. Compute the statistic on the resample.
  4. Repeat many times.
  5. Use the resulting distribution to estimate standard errors or intervals.

Bootstrap methods do not create new population information. They use the empirical sample as a stand-in for the unknown population distribution.

Sampling Distribution versus Bootstrap Distribution

The true sampling distribution is generated by repeated samples from the population or probability model.

A bootstrap distribution is generated by repeated resamples from the observed data.

The bootstrap is an approximation strategy, not a literal repetition of the original population-sampling process.

Common Error: Confusing Observations with Statistics

A histogram of \(500\) individual observations is not automatically a sampling distribution.

A sampling distribution requires repeated values of a statistic.

For example, \(500\) sample means from \(500\) repeated samples could form an empirical approximation to a sampling distribution.

Common Error: Thinking One Sample Reveals the Sampling Distribution

One observed sample gives one observed value of the statistic.

The sampling distribution is a theoretical or simulated repeated-sampling object.

Its properties come from the model, the design, mathematical derivation, or repeated simulation—not from a single observed statistic alone.

Common Error: Using \(\sigma\) and Standard Error Interchangeably

For the sample mean,

\[ \sigma \]

describes the standard deviation of individual population observations, while

\[ \frac{\sigma}{\sqrt{n}} \]

describes the standard deviation of sample means.

The second quantity is the standard error of \(\bar{X}\).

Common Error: Assuming Larger Samples Remove Bias

Increasing \(n\) generally reduces sampling variability under an appropriate probability model.

It does not automatically remove systematic bias caused by a poor sampling frame, nonresponse, measurement error, or a flawed estimator.

A tightly concentrated sampling distribution can still be centered at the wrong value.

Common Error: Treating \(n\ge 30\) as a Universal Law

The statement “the sampling distribution is normal whenever \(n\ge 30\)” is too strong.

Approximation quality depends on the population distribution, the statistic, and the sampling structure.

Thirty observations may be more than sufficient in some settings and insufficient in others.

Common Error: Ignoring Dependence

Many standard sampling-distribution formulas assume independent or approximately independent observations.

Repeated measurements on the same person, clustered observations, time series, spatial data, and network data can violate that assumption.

Dependence changes variance calculations and can make naive standard errors too small or otherwise incorrect.

A Reliable Procedure for Analyzing a Sampling Distribution

  1. Identify the population parameter of interest.
  2. Identify the statistic used to estimate it.
  3. State the sampling design or probability model.
  4. Determine the center of the statistic's sampling distribution.
  5. Determine or estimate its standard error.
  6. Determine the shape exactly or through an approximation.
  7. Check independence and design conditions.
  8. Check approximation conditions when using a normal or other reference distribution.
  9. Apply finite-population corrections when required.
  10. Interpret results in terms of repeated sampling.

Example: Full Sample-Mean Workflow

Suppose a population has

\[ \mu=120,\qquad \sigma=24, \]

and random samples of size

\[ n=64 \]

are taken.

The center of the sampling distribution is

\[ E(\bar{X})=120. \]

The standard error is

\[ \operatorname{SE}(\bar{X}) = \frac{24}{\sqrt{64}} = 3. \]

If the population is normal, or if a normal approximation is justified,

\[ \bar{X} \approx N(120,3^2). \]

A sample mean of \(126\) is

\[ z=\frac{126-120}{3}=2 \]

standard errors above the population mean.

Example: Full Sample-Proportion Workflow

Suppose a population proportion is

\[ p=0.30 \]

and a random sample has size

\[ n=500. \]

The center is

\[ E(\hat{p})=0.30. \]

The standard error is

\[ \operatorname{SE}(\hat{p}) = \sqrt{\frac{0.30(0.70)}{500}} \approx 0.0205. \]

The success-failure counts are

\[ np=150,\qquad n(1-p)=350, \]

so a normal approximation is generally reasonable.

Practice 1: Identify the Sampling Distribution

A researcher repeatedly draws random samples of \(50\) households and calculates the average monthly electricity use for each sample.

What distribution is formed by the collection of sample averages?

Practice 2: Standard Error of a Mean

A population has standard deviation

\[ \sigma=20. \]

Find the standard error of the sample mean for

\[ n=100. \]

Practice 3: Effect of Sample Size

If the standard error of a sample mean is \(8\) when the sample size is \(n\), what will the standard error be when the sample size is increased to \(4n\)?

Practice 4: Standard Error of a Proportion

Suppose

\[ p=0.64,\qquad n=400. \]

Find the standard error of \(\hat{p}\).

Practice 5: Standardize a Sample Mean

Suppose

\[ \mu=80,\qquad \sigma=16,\qquad n=64. \]

A sample has mean

\[ \bar{x}=84. \]

How many standard errors above the population mean is the sample mean?

Practice 6: Check a Normal Approximation

Suppose

\[ p=0.08,\qquad n=50. \]

Using the common success-failure check, is a normal approximation for \(\hat{p}\) well supported?

Practice 7: Finite Population Correction

A simple random sample of \(n=100\) is selected without replacement from a population of \(N=500\).

Find the finite population correction factor.

Practice 8: Identify the Error

A student says, “The sample standard deviation is \(12\), so the standard error of the sample mean must also be \(12\).”

Explain the error.

Practice 9: Sampling Distribution versus Population Distribution

A population is strongly right-skewed. Large random samples are repeatedly drawn and their means are recorded.

Does the central limit theorem say the original population becomes normal?

Practice 10: Bias versus Variability

An estimator has a very narrow sampling distribution, but the distribution is centered well above the true parameter.

Is the main problem high variability, bias, or both?

Practice Answers

Practice 1

The values form the sampling distribution of the sample mean.

Practice 2

\[ \operatorname{SE}(\bar{X}) = \frac{20}{\sqrt{100}} = \boxed{2}. \]

Practice 3

Multiplying the sample size by \(4\) divides the standard error by \(\sqrt{4}=2\).

\[ \boxed{4}. \]

Practice 4

\[ \operatorname{SE}(\hat{p}) = \sqrt{\frac{0.64(0.36)}{400}} = \sqrt{0.000576} = \boxed{0.024}. \]

Practice 5

First,

\[ \operatorname{SE}(\bar{X}) = \frac{16}{\sqrt{64}} = 2. \]

Therefore,

\[ z=\frac{84-80}{2} = \boxed{2}. \]

The sample mean is two standard errors above the population mean.

Practice 6

\[ np=50(0.08)=4, \] \[ n(1-p)=50(0.92)=46. \]

Because \(np=4\) is below the common threshold of \(10\), the usual normal approximation is not well supported by that rule.

Practice 7

\[ \sqrt{\frac{N-n}{N-1}} = \sqrt{\frac{500-100}{499}} = \sqrt{\frac{400}{499}} \approx \boxed{0.895}. \]

Practice 8

The sample standard deviation measures variability among observations. The estimated standard error of the mean is

\[ \frac{s}{\sqrt{n}}, \]

not \(s\) itself.

Practice 9

No.

The central limit theorem concerns the sampling distribution of the standardized sample mean. It does not transform the original population distribution into a normal distribution.

Practice 10

The main problem is bias.

The narrow distribution indicates relatively low variability, but its center is systematically displaced from the true parameter.

What to Remember

A sampling distribution describes how a statistic behaves across repeated random samples.

The most important habit is: separate the variability of individual observations from the variability of statistics across samples.

Where We Go Next

Sampling distributions give us a framework for studying the center and spread of estimators, but good inference also depends on whether the sampling process systematically favors some outcomes over others and whether the observed sample adequately reflects the population of interest.

The next lesson, Bias, Variability, and Representativeness, compares these dimensions directly and develops a practical framework for evaluating the quality of statistical estimates.