The Central Idea

Suppose a population contains many units, but measuring every unit is impractical. We select a smaller collection and use it to learn about the whole.

The central question is not merely How many observations did we collect?

It is also How were those observations selected?

A random sample is produced by a sampling mechanism in which selection is governed by chance according to a specified probability rule.

Randomness does not guarantee that every sample will perfectly resemble the population. It provides a principled mechanism for controlling selection and for describing the variability that remains from sample to sample.

Population and Sample

The population is the complete collection of units about which we want information.

A sample is the subset of units actually observed.

If a university wants to estimate the average weekly study time of all \(12{,}000\) enrolled students, then the population is the full set of \(12{,}000\) students.

If \(400\) students are selected and surveyed, those \(400\) students form the sample.

We often denote population size by

\[ N \]

and sample size by

\[ n. \]

Units, Cases, and Observational Units

A sampling unit is an object that can be selected during the sampling process.

Depending on the study, a sampling unit could be:

Clearly identifying the unit matters because the probability of selection is defined at the level of the unit.

Parameters and Statistics

A parameter is a numerical characteristic of a population.

A statistic is a numerical characteristic computed from a sample.

For example, the population mean is often denoted

\[ \mu, \]

while the sample mean is

\[ \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i. \]

Statistical inference uses statistics such as \(\bar{x}\) to learn about unknown parameters such as \(\mu\).

Why We Sample

A complete census may be expensive, slow, impossible, or unnecessary.

Sampling can reduce:

A carefully designed sample can therefore provide useful information about a large population without observing every unit.

Sampling Is Not the Same as Convenience

Selecting whoever happens to be easiest to reach is called convenience sampling.

For example, asking only students sitting near the entrance of a library about campus study habits may be easy, but those students may differ systematically from the full student population.

Convenience samples can be useful for exploratory work, but they do not automatically support population-level inference.

Random Selection

Random selection means that chance determines which units enter the sample according to a known selection rule.

The rule might assign equal probability to every unit, or it might assign different known probabilities as part of a more complex design.

The key feature is that selection is not left entirely to interviewer choice, volunteer choice, ease of access, or another uncontrolled mechanism.

Simple Random Sampling

A simple random sample, or SRS, of size \(n\) from a population of size \(N\) is a sample selected so that every possible subset of \(n\) distinct population units has the same probability of being chosen.

The number of possible samples is

\[ \binom{N}{n}. \]

Therefore, under simple random sampling, each possible sample has probability

\[ \boxed{\frac{1}{\binom{N}{n}}}. \]

Example: A Small Simple Random Sample

Suppose a group contains five people:

\[ A,\quad B,\quad C,\quad D,\quad E. \]

We want a simple random sample of size \(2\).

The number of possible samples is

\[ \binom{5}{2}=10. \]

Therefore, each pair must have probability

\[ \frac{1}{10} \]

of being selected.

Equal Inclusion Probability in an SRS

Under a simple random sample of size \(n\) from \(N\) units, each individual unit has the same probability of inclusion.

That probability is

\[ \boxed{\frac{n}{N}}. \]

For example, if \(100\) units are sampled from a population of \(2{,}000\), then each unit has inclusion probability

\[ \frac{100}{2000}=0.05. \]

Each unit therefore has a \(5\%\) chance of appearing in the sample.

Equal Individual Chances Are Not the Whole Definition

It is important to distinguish equal individual inclusion probabilities from a full simple random sample.

A design can give every individual the same probability of selection while still making some combinations of individuals impossible.

A simple random sample requires equal probability for every possible sample of the specified size.

Sampling with and without Replacement

In sampling without replacement, once a unit is selected, it cannot be selected again.

In sampling with replacement, a selected unit is returned to the pool and can be selected again.

Most surveys of distinct people use sampling without replacement because the goal is usually to observe different people.

Sampling with replacement is common in theoretical probability and in computational resampling methods.

How to Draw a Simple Random Sample

A practical procedure is:

  1. Define the population.
  2. Create or obtain a sampling frame containing the eligible units.
  3. Assign a unique identifier to each unit.
  4. Use a random mechanism to select the required number of identifiers.
  5. Observe the corresponding units.
  6. Record nonresponse and other departures from the intended design.

The random mechanism may be a physical lottery, a random-number generator, or another verified probability procedure.

The Sampling Frame

A sampling frame is the operational list or mechanism from which units are selected.

Examples include:

The sampling frame should align as closely as possible with the target population.

Target Population versus Frame Population

The target population is the population we want to describe.

The frame population is the set of units that can actually be reached through the sampling frame.

If some members of the target population are missing from the frame, the design has undercoverage.

If the frame contains units that do not belong to the target population, it has overcoverage.

Example: Undercoverage

Suppose a city wants to estimate transportation habits for all adult residents but selects names only from a driver's-license registry.

Adults without driver's licenses cannot be selected from that frame.

The resulting problem is not ordinary random fluctuation. It is a mismatch between the target population and the selection mechanism.

Sampling Error

Even a well-designed random sample will usually differ somewhat from the population.

The difference between a sample statistic and the corresponding population parameter is called sampling error.

For a sample mean,

\[ \text{sampling error}=\bar{x}-\mu. \]

The term error here does not necessarily mean a mistake. It refers to the random difference caused by observing only part of the population.

Random Sampling Produces Sampling Variability

If we repeatedly take random samples from the same population, the sample statistic will generally change from one sample to another.

This phenomenon is called sampling variability.

Random sampling does not eliminate variability. Instead, it creates a probability structure that allows that variability to be studied.

The next lesson will formalize this idea through sampling distributions.

Random Sampling and Representativeness

A random sample is often described informally as being intended to represent the population.

But randomness does not force every sample to match the population on every characteristic.

By chance, one random sample may contain more young people than the population proportion, another may contain fewer, and a third may be very close.

The benefit of random sampling is that these differences arise from a controlled probability mechanism rather than from systematic human selection.

Larger Random Samples Tend to Be More Stable

Other things being equal, statistics calculated from larger random samples tend to vary less from sample to sample than statistics calculated from smaller random samples.

For many common estimators, variability decreases on the order of

\[ \frac{1}{\sqrt{n}}. \]

This does not mean that doubling the sample size cuts uncertainty in half.

To reduce a quantity proportional to \(1/\sqrt{n}\) by half, the sample size must be multiplied by approximately \(4\).

Random Sampling Does Not Repair a Bad Frame

A perfectly randomized selection from an incomplete sampling frame is still limited by that frame.

If a telephone survey randomly samples only landline numbers, then people reachable only by mobile phone are excluded.

The randomization can be correct while the coverage remains inadequate.

Random Sampling Does Not Eliminate Nonresponse

A selected unit may refuse to participate, fail to respond, or be unreachable.

This creates nonresponse.

If responders and nonresponders differ systematically on characteristics related to the study, nonresponse can distort the observed sample.

The original random selection is still valuable, but follow-up procedures and analysis must account for what happened after selection.

Random Sampling Does Not Eliminate Measurement Error

A sample can be selected perfectly and still produce poor data if the measurements themselves are inaccurate.

Measurement error can arise from:

Sampling design and measurement design solve different problems.

Probability Sampling

A probability sample is selected by a design in which the probabilities governing selection are known or specified.

Simple random sampling is one probability-sampling design, but it is not the only one.

Other important probability designs include:

Systematic Random Sampling

In systematic sampling, we select a random starting position and then take units at regular intervals through an ordered frame.

If a list contains \(N=10{,}000\) units and we want approximately \(n=500\), then the sampling interval is about

\[ k=\frac{N}{n}=\frac{10{,}000}{500}=20. \]

We might choose a random starting position from \(1\) through \(20\), then select every \(20\)th unit.

A Caution with Systematic Sampling

Systematic sampling can be efficient, but the ordering of the frame matters.

If the list contains a repeating pattern that aligns with the sampling interval, the resulting sample can be distorted.

For example, selecting every seventh record can be problematic if the records themselves follow a seven-unit cycle related to the variable of interest.

Stratified Random Sampling

In stratified random sampling, the population is divided into nonoverlapping groups called strata, and a random sample is selected within each stratum.

Common strata might be based on:

Stratification can ensure that important subgroups are represented in the sample.

Proportional Stratified Sampling

In a proportional allocation, the sample from each stratum is chosen in proportion to that stratum's share of the population.

If stratum \(h\) has population size \(N_h\), a proportional sample size is approximately

\[ n_h=n\frac{N_h}{N}. \]

Suppose a population is \(60\%\) Group A and \(40\%\) Group B, and we want a sample of \(500\).

A proportional allocation would use approximately

\[ 500(0.60)=300 \]

from Group A and

\[ 500(0.40)=200 \]

from Group B.

Disproportionate Stratified Sampling

Sometimes a small subgroup is scientifically or operationally important but would contribute too few observations under proportional sampling.

A study may deliberately sample that stratum at a higher rate.

This is called disproportionate stratified sampling.

Because different units may then have different inclusion probabilities, analysis often uses sampling weights to reconstruct population-level quantities.

Cluster Sampling

In cluster sampling, the population is divided into natural groups called clusters.

We randomly select clusters and then observe units within the selected clusters.

Examples of clusters include:

Cluster sampling is often used when creating a complete list of all individual units would be difficult or expensive.

Strata and Clusters Serve Different Purposes

Stratified sampling and cluster sampling both divide a population into groups, but their roles differ.

In stratified sampling, we usually sample from every stratum.

In cluster sampling, we usually select only some clusters.

Stratification is often used to improve subgroup coverage or precision. Clustering is often used to make data collection more practical.

Multistage Sampling

Large surveys often select units in several stages.

For example:

  1. randomly select counties;
  2. within selected counties, randomly select neighborhoods;
  3. within selected neighborhoods, randomly select households;
  4. within selected households, randomly select individuals.

This is a multistage sampling design.

The resulting probabilities can be more complicated than those of a simple random sample, but each stage can still be governed by a probability rule.

Sampling Weights

When different units have different probabilities of inclusion, sampled units may be assigned sampling weights.

A basic design weight is the reciprocal of the unit's inclusion probability:

\[ \boxed{w_i=\frac{1}{\pi_i}}, \]

where \(\pi_i\) is the probability that unit \(i\) enters the sample.

If a unit has inclusion probability \(0.02\), then its basic design weight is

\[ \frac{1}{0.02}=50. \]

Informally, that sampled unit represents about \(50\) population units under the design.

Self-Selected and Voluntary-Response Samples

A voluntary-response sample is formed when people choose for themselves whether to participate.

Examples include:

The people who volunteer may differ systematically from those who do not.

Large sample size does not by itself remove this selection problem.

Random Sample versus Random Assignment

Random sampling and random assignment are different ideas.

Random sampling concerns who enters the study.

Random assignment concerns which treatment or condition a study participant receives.

Random sampling supports population generalization when the sampling design is appropriate.

Random assignment supports causal comparisons by balancing treatment groups through chance.

A Study Can Use One Without the Other

A researcher could recruit volunteers and then randomly assign them to two treatments.

That study uses random assignment but not random sampling.

Conversely, a researcher could take a random sample of households and simply observe their current energy use.

That study uses random sampling but no random assignment.

The two forms of randomization address different inferential questions.

Finite Population Correction

When sampling without replacement from a finite population, taking a large fraction of the population reduces uncertainty because fewer unobserved units remain.

For a simple random sample, a common finite population correction is

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

When \(n\) is very small relative to \(N\), this factor is close to \(1\).

When a substantial fraction of the population is sampled, the correction can matter.

The Sampling Fraction

The sampling fraction is

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

If \(n=500\) and \(N=20{,}000\), then

\[ f=\frac{500}{20{,}000}=0.025. \]

The sample contains \(2.5\%\) of the population.

Random Number Generation

Modern random sampling is often implemented with software.

A basic workflow is:

  1. assign each unit a unique index;
  2. generate random indices according to the design;
  3. remove duplicates when sampling without replacement;
  4. retain the selected units;
  5. store the random seed when reproducibility is needed.

The computational tool does not replace the sampling design. It implements the design.

Reproducibility and Random Seeds

A pseudorandom number generator creates a deterministic sequence from an initial value called a seed.

Recording the seed allows the same random sequence to be reproduced later.

This can be useful for:

Example: Selecting Student IDs

Suppose a university has \(8{,}000\) eligible students and wants a simple random sample of \(200\).

Give each student a unique identifier from \(1\) to \(8{,}000\).

Use a random-number generator to select \(200\) distinct identifiers.

Each student has inclusion probability

\[ \frac{200}{8000}=0.025. \]

Thus each eligible student has a \(2.5\%\) chance of entering the sample.

Example: Stratifying by Campus

Suppose a university system has three campuses:

The total population is

\[ N=10{,}000. \]

For a proportional stratified sample of \(n=500\):

\[ n_A=500\left(\frac{5000}{10000}\right)=250, \] \[ n_B=500\left(\frac{3000}{10000}\right)=150, \] \[ n_C=500\left(\frac{2000}{10000}\right)=100. \]

Then a random sample is drawn separately within each campus.

Example: Cluster Sampling Schools

Suppose a district contains \(80\) schools and researchers want information about student cafeteria use.

Instead of sampling individual students from the entire district, the researchers might randomly select \(10\) schools and then survey students within those schools.

The schools act as clusters.

This can reduce travel and administrative cost, although students within the same school may be more similar to one another than students selected independently across the district.

Design Effect

Complex sampling designs can produce more or less variability than a simple random sample of the same size.

A common comparison is the design effect:

\[ \boxed{\operatorname{DEFF} =\frac{\operatorname{Var}(\text{estimator under actual design})} {\operatorname{Var}(\text{estimator under SRS})}}. \]

A design effect greater than \(1\) indicates more variance than the SRS benchmark for that estimator and sample size.

A design effect below \(1\) indicates greater precision than that benchmark.

Why Cluster Samples Can Increase Variability

Units in the same cluster often resemble one another.

Students in the same classroom share a teacher, schedule, and environment. Households on the same block share local conditions.

If many sampled observations contain similar information, the effective amount of independent information can be smaller than the raw sample size suggests.

Why Stratification Can Reduce Variability

Stratification can improve precision when units within each stratum are relatively similar and the strata differ meaningfully from one another.

By controlling how many observations come from each stratum, the design can reduce chance imbalance in important population composition.

The benefit depends on how the strata are defined and how the estimator is constructed.

Sample Size and Population Size

A common misconception is that a large population always requires a sample that is a large percentage of the population.

For many estimation problems, the absolute sample size \(n\) matters more directly than the population percentage once the population is large relative to the sample.

A random sample of \(1{,}000\) can provide substantial information about a population of \(100{,}000\) or \(10{,}000{,}000\), provided the design and measurement process are appropriate.

The next lessons will quantify this through standard errors and sampling distributions.

Selection Probability Must Match the Intended Design

Calling a sample “random” is not enough.

A defensible design should make clear:

Randomness is a property of a procedure, not a label attached afterward.

Common Error: Choosing a Sample That Merely Looks Representative

Suppose a population is \(60\%\) one category and \(40\%\) another.

A researcher might manually construct a sample with exactly those percentages and call it representative.

Matching one known characteristic does not guarantee that the selection is unbiased with respect to other characteristics.

Probability sampling controls the selection mechanism rather than relying only on visual or demographic resemblance.

Common Error: Assuming Random Means Perfectly Balanced

A random process can produce imbalance.

If a population is half Category A and half Category B, a random sample of \(20\) does not have to contain exactly \(10\) from each category.

Samples such as \(9\) and \(11\), or \(12\) and \(8\), can occur naturally.

Randomness controls probability, not exact composition.

Common Error: Confusing Sample Size with Sampling Quality

A very large biased sample can still produce a misleading estimate.

For example, an online poll with \(100{,}000\) voluntary responses may have more observations than a carefully selected probability sample of \(2{,}000\), yet the larger poll can suffer severe self-selection bias.

More observations reduce random sampling variability only when the underlying data-collection process is sufficiently appropriate for the inferential goal.

Common Error: Confusing Random Sampling with Random Assignment

Selecting \(200\) people randomly from a population is random sampling.

Taking those \(200\) people and randomly assigning \(100\) to Treatment A and \(100\) to Treatment B is random assignment.

The first concerns population selection. The second concerns treatment allocation.

Common Error: Ignoring the Sampling Frame

Suppose a business wants to measure satisfaction among all customers but samples randomly only from customers enrolled in its rewards program.

The selection can be random within the rewards database while still missing all nonmember customers.

The sampling frame must be evaluated separately from the randomization method.

Common Error: Treating Nonresponse as Random Without Evidence

If \(1{,}000\) people are randomly selected but only \(250\) respond, the final respondent set is not automatically equivalent to a random sample of \(250\).

The response mechanism may depend on age, time availability, interest, income, or other factors.

Nonresponse must therefore be examined rather than silently treated as harmless random loss.

A Reliable Procedure for Designing a Random Sample

  1. State the target population precisely.
  2. Define the sampling unit.
  3. Identify or construct the sampling frame.
  4. Check the frame for undercoverage, overcoverage, and duplication.
  5. Choose an appropriate probability-sampling design.
  6. Determine the intended sample size.
  7. Specify inclusion probabilities.
  8. Implement the random selection exactly as designed.
  9. Track contact attempts, nonresponse, and substitutions if any.
  10. Document departures from the design.
  11. Use weights when the design requires them.
  12. Preserve enough information to reproduce or audit the selection.

Example: Full Simple-Random-Sample Workflow

A company has \(N=4{,}800\) active customers and wants a sample of \(n=240\).

The company first creates a frame containing one record for each eligible active customer.

It assigns IDs \(1\) through \(4{,}800\) and selects \(240\) distinct IDs by a random-number generator.

Each customer has inclusion probability

\[ \pi_i=\frac{240}{4800}=0.05. \]

Thus each eligible customer has a \(5\%\) chance of being selected.

If some selected customers do not respond, the company records those outcomes rather than silently replacing them with conveniently available customers.

Example: Full Stratified Workflow

Suppose a population contains \(12{,}000\) employees:

For a proportional sample of \(600\):

\[ n_{\text{Operations}} =600\left(\frac{6000}{12000}\right)=300, \] \[ n_{\text{Sales}} =600\left(\frac{3000}{12000}\right)=150, \] \[ n_{\text{Technology}} =600\left(\frac{2000}{12000}\right)=100, \] \[ n_{\text{Administration}} =600\left(\frac{1000}{12000}\right)=50. \]

Each stratum then receives its own random selection.

Practice 1: Population and Sample

A county has \(85{,}000\) registered voters. Researchers randomly select \(1{,}200\) of them for a survey.

Identify the population and the sample.

Practice 2: Inclusion Probability

A simple random sample of \(300\) units is selected from a population of \(7{,}500\).

What is the inclusion probability of each unit?

Practice 3: Number of Possible Samples

A population contains \(8\) units and a simple random sample of \(3\) is selected.

How many different samples are possible?

Practice 4: Identify the Sampling Method

A company chooses a random starting employee from the first \(25\) names on an alphabetized roster and then selects every \(25\)th employee.

What sampling method is being used?

Practice 5: Proportional Stratification

A school district has \(9{,}000\) students:

A proportional stratified sample of \(600\) students is planned.

How many students should be selected from each stratum?

Practice 6: Cluster or Stratum?

A researcher randomly selects \(12\) apartment buildings from a city and surveys all households in those selected buildings.

Is this primarily stratified sampling or cluster sampling?

Practice 7: Sampling Fraction

A sample contains \(400\) units from a population of \(16{,}000\).

Find the sampling fraction.

Practice 8: Identify the Design Problem

A university wants to study all enrolled students but randomly samples only from students who have purchased a campus parking permit.

What major sampling problem should be investigated?

Practice 9: Sampling or Assignment?

Researchers randomly choose \(500\) households from a city directory and then observe their current recycling behavior.

Is the randomization here random sampling, random assignment, or both?

Practice 10: Explain the Error

A website receives \(75{,}000\) voluntary responses to a poll. A student claims the result must accurately estimate the entire population because the sample is extremely large.

Explain the flaw in the reasoning.

Practice Answers

Practice 1

The population is all \(85{,}000\) registered voters in the county.

The sample is the \(1{,}200\) voters selected for the survey.

Practice 2

Under simple random sampling,

\[ \pi_i=\frac{n}{N} =\frac{300}{7500} =0.04. \]

Each unit has a \(4\%\) probability of inclusion.

Practice 3

\[ \binom{8}{3} =\frac{8!}{3!5!} =56. \]

There are

\[ \boxed{56} \]

possible samples.

Practice 4

This is systematic random sampling.

A random starting point is chosen, followed by selection at a fixed interval.

Practice 5

Elementary:

\[ 600\left(\frac{4500}{9000}\right)=300. \]

Middle school:

\[ 600\left(\frac{3000}{9000}\right)=200. \]

High school:

\[ 600\left(\frac{1500}{9000}\right)=100. \]

The proportional allocation is therefore

\[ \boxed{300,\quad 200,\quad 100}. \]

Practice 6

This is cluster sampling.

Entire apartment buildings are selected as groups, and households are observed within the selected groups.

Practice 7

\[ f=\frac{n}{N} =\frac{400}{16000} =0.025. \]

The sampling fraction is

\[ \boxed{0.025=2.5\%}. \]

Practice 8

The major issue is undercoverage.

Students without parking permits cannot be selected from the frame even though they belong to the target population.

Practice 9

The study uses random sampling.

Households are randomly selected from the population frame, but no treatment is assigned, so there is no random assignment.

Practice 10

The sample is voluntary rather than probability-selected.

People who choose to respond may differ systematically from those who do not. Increasing the number of voluntary responses can reduce some random fluctuation, but it does not automatically remove self-selection bias.

What to Remember

Random sampling gives probability a formal role in deciding which units enter a sample.

The most important habit is: evaluate the selection process before interpreting the sample statistic.

Where We Go Next

A random sample is not fixed in advance by the population. If we repeated the sampling process, we would usually obtain different observations and a different value of the sample statistic.

The next lesson, Sampling Distributions, studies the probability distributions generated by statistics across repeated random samples and provides the foundation for standard errors, confidence intervals, and hypothesis tests.