Sampling and Sampling Distributions

Introduction

Business analysts rarely have the time, money, or access required to study every member of a population. A retailer may want to understand all customers, a bank may wish to estimate the default rate of all borrowers, and a government agency may need to measure the employment status of a large population. In each case, analysis is usually based on a sample: a smaller group selected from the population.

Sampling is not simply a cost-saving technique. The method used to select a sample determines the credibility of the conclusions drawn from it. A carefully designed probability sample can support statistical generalisation, whereas a convenient but biased sample may produce precise-looking results that are systematically wrong. Statistical sampling also provides the foundation for confidence intervals, hypothesis tests, surveys, quality control, market research and predictive analytics.

This chapter explains probability and non-probability sampling, sample-size determination, sampling error, the central limit theorem and sampling distributions. It also shows how these concepts guide practical business decisions.

Learning Objectives

By the end of this chapter, you should be able to:

1. Distinguish populations, samples, parameters and statistics.
2. Compare probability and non-probability sampling methods.
3. Select an appropriate sampling method for a business problem.
4. Determine a sample size for estimating a population proportion or mean.
5. Explain sampling error, bias and nonsampling error.
6. Describe sampling distributions and apply the central limit theorem.
7. Calculate and interpret standard errors for sample means and proportions.
8. Evaluate sampling decisions ethically and professionally.

1. Foundations of Sampling

A population is the complete set of units relevant to a study. A unit may be a person, transaction, household, product, store or company. The target population is the population to which the researcher intends to generalise findings. A sampling frame is the operational list or database from which the sample is selected. The sampling frame should correspond closely to the target population, although in practice it may omit some units or include ineligible ones.

A sample is a subset of the population. A numerical characteristic of a population is a parameter, while a numerical characteristic calculated from a sample is a statistic.

The purpose of inferential statistics is to use sample statistics to estimate unknown population parameters. This requires two conditions: the sample must be relevant to the target population, and the uncertainty caused by sampling must be quantified.

Census and sample

A census observes every population unit. It eliminates sampling error but can be expensive, slow and impractical. A sample is usually preferable when the population is large, measurements are destructive, or a timely decision is required. Sampling is not automatically inferior to a census: a well-designed sample may produce more reliable results than a poorly managed census because fewer observations can be checked carefully.

2. Probability Sampling

In probability sampling, every population unit has a known, non-zero probability of selection. Randomisation is used to select units, making it possible to estimate sampling error and generalise results under appropriate assumptions.

Simple random sampling

In simple random sampling, every possible sample of a specified size has an equal chance of selection. A random-number generator can select customer identification numbers from a complete sampling frame.

Advantages include conceptual simplicity and unbiased selection when the frame is complete. Limitations include the need for a reliable population list and the possibility that a purely random sample does not adequately represent small subgroups.

Systematic sampling

In systematic sampling, the researcher selects every kth unit after choosing a random starting point. If the frame contains N units and the desired sample size is n, the approximate sampling interval is:

Here, N is the frame size, n is the sample size and k is the interval. For example, if N = 12,000 customers and n = 400, then k = 30. After a random start between 1 and 30, every thirtieth customer is selected.

Systematic sampling is efficient, but it can be biased if the list contains a periodic pattern matching k. For example, selecting every tenth production item may repeatedly select the same shift or machine position.

Stratified sampling

In stratified sampling, the population is divided into non-overlapping subgroups called strata, and a probability sample is selected from each stratum. Strata may be based on region, customer type, age group or business size.

Proportional stratified sampling gives each stratum a sample share equal to its population share. If stratum h has population size Nₕ and the total population is N, its proportional sample size is:

Stratification is useful when groups differ substantially and each group must be represented. It can improve precision compared with a simple random sample of the same size. A difficulty is that accurate information is needed to classify units before sampling.

Cluster sampling

In cluster sampling, the population is divided into naturally occurring groups, such as branches, schools, stores or geographic areas. A random sample of clusters is selected, and all or some units within selected clusters are studied.

Cluster sampling reduces travel and administration costs. However, units within a cluster are often similar, so a given number of observations may provide less information than observations spread across many clusters. This loss of efficiency is commonly described through the design effect. If a cluster sample has substantial within-cluster similarity, its effective sample size is smaller than its actual number of observations.

Multistage sampling

Multistage sampling combines stages. For example, a national business survey might randomly select regions, then firms within regions, then employees within firms. It is practical for large and geographically dispersed populations, but its design and analysis are more complex. Analysts should account for unequal selection probabilities and clustering when estimating uncertainty.

3. Non-Probability Sampling

In non-probability sampling, the probability that each population unit is selected is unknown. These methods may be appropriate for exploratory research, qualitative studies, pilot testing or situations where no adequate sampling frame exists. However, conventional margins of error should not be presented as though the sample were a probability sample.

A large non-probability sample is not necessarily representative. Increasing sample size mainly reduces random variation; it does not correct systematic bias. For example, surveying 10,000 highly dissatisfied customers who voluntarily respond may produce a stable but unrepresentative estimate of all customers.

Choosing between methods

The choice should reflect the research objective, population structure, resources, required precision and ethical constraints. Probability sampling is preferred when the goal is population estimation and generalisation. Non-probability methods may be defensible when the goal is discovery, when the population is difficult to identify, or when the study is an early-stage investigation. The method and its limitations must be reported transparently.

4. Sampling Error and Nonsampling Error

Sampling error is the difference between a sample statistic and the corresponding population parameter that arises because only part of the population is observed. It varies from sample to sample. Even a properly selected random sample can have a sample mean different from μ.

Sampling error is not the same as a mistake. It is an expected consequence of sampling and can be estimated probabilistically. It generally decreases as sample size increases, although it does not normally disappear entirely.

Nonsampling errors arise from problems other than selecting a sample. Important examples include:

• Coverage error: the sampling frame omits eligible units or includes ineligible ones.
• Nonresponse error: selected units do not respond, and respondents differ systematically from nonrespondents.
• Measurement error: questions, instruments or recording procedures produce inaccurate values.
• Processing error: data are incorrectly coded, entered, merged or analysed.
• Interviewer error: the interviewer influences responses or records them inaccurately.

Increasing sample size does not automatically reduce these errors. Effective quality control includes frame assessment, clear questionnaire design, interviewer training, follow-up of nonrespondents, validation checks and documentation of exclusions.

5. Sampling Distributions

A sampling distribution is the probability distribution of a statistic over all possible random samples of a specified size from a population. It is a theoretical distribution, although it can be approximated through repeated sampling or simulation.

The sampling distribution explains how much a statistic is expected to vary. Its centre describes typical performance, and its spread is the standard error.

Sampling distribution of the sample mean

Suppose independent observations are drawn from a population with mean μ and finite standard deviation σ. For a sample of size n, the sample mean is:

The expected value and standard error of the sample mean are:

If σ is unknown, it is commonly estimated by:

The standard error is measured in the same units as the original variable. It is not the same as the standard deviation of individual observations. As n increases, the standard error decreases in proportion to 1 ÷ √n. Therefore, quadrupling the sample size approximately halves the standard error.

If the population is finite and sampling is without replacement, and the sample is a substantial fraction of the population, a finite population correction may be appropriate:

Here N is the population size. The correction is close to 1 when the sample is small relative to the population and is often considered when the sampling fraction n ÷ N is greater than about 5%.

Sampling distribution of a sample proportion

For a binary characteristic, let p be the population proportion and p̂ the sample proportion. If n independent observations are taken under suitable conditions, then:

Because p is usually unknown, analysts may use p̂ in its place:

The normal approximation is generally reasonable when np and n(1 − p) are both sufficiently large, commonly at least 10. For small counts, exact or alternative methods may be more appropriate.

6. The Central Limit Theorem

The central limit theorem (CLT) states that, under broad conditions, the sampling distribution of the sample mean becomes approximately normal as sample size increases, even when the population distribution is not normal. Its centre is μ and its standard deviation is σ ÷ √n.

The CLT does not claim that the original data become normal. It concerns the distribution of sample means across repeated samples. The required sample size depends on the population shape: moderately skewed populations may require a larger sample, while extremely skewed or heavy-tailed populations may require much larger samples or robust methods.

If the population itself is normal, the sampling distribution of x̄ is normal for every sample size when observations are independent. Independence may be threatened by repeated observations from the same customer, time-series dependence, household clustering or organisational grouping.

For a normal or approximately normal sampling distribution, a standardised sample mean is:

When σ is unknown and the population is approximately normal, the t-statistic is used:

The t distribution has n − 1 degrees of freedom. It has heavier tails than the standard normal distribution, reflecting extra uncertainty from estimating σ with s. As n becomes large, the t distribution approaches the standard normal distribution.

7. Determining Sample Size

Sample-size planning should occur before data collection. It requires a decision about confidence, precision, variability and available resources. A larger sample is not always necessary; it should be large enough to support the intended decision.

Sample size for a population proportion

For estimating a population proportion with a desired margin of error E at a confidence level associated with critical value z, the basic planning formula is:

Here n₀ is the initial required sample size, p* is the anticipated population proportion, E is the maximum desired margin of error expressed as a proportion, and z is the standard-normal critical value. Common values are 1.645 for 90% confidence, 1.96 for 95% confidence and 2.576 for 99% confidence.

If no prior estimate of p is available, p = 0.5 is conservative because p(1 − p*) is then largest. The formula assumes a probability sample, independent observations or an appropriate design adjustment, and a sufficiently large sample for the chosen interval method.

For a finite population, a corrected sample size is:

where N is the population size.

Sample-size example: proportion

A firm wants to estimate the proportion of customers who would recommend its service. It seeks 95% confidence and a margin of error of ±0.05. No prior estimate is available, so p* = 0.5.

The required sample is rounded upward to 385 customers. If the population contains only N = 2,000 customers, the finite-population correction gives:

Thus, at least 323 completed responses would be required under the stated assumptions. If nonresponse is expected, more customers must be contacted. For an anticipated response rate r, the number contacted is approximately n ÷ r. At a 70% response rate, contacting 323 ÷ 0.70 ≈ 462 customers would be prudent.

Sample size for a population mean

When the objective is to estimate a population mean with margin of error E, an initial formula is:

Here σ is the anticipated population standard deviation in the measurement units. If σ is unknown, it may be estimated from a pilot study, historical data or a comparable study. The formula assumes a probability sample, an appropriate distributional approximation and a meaningful definition of E.

For example, if a delivery company expects a standard deviation of 12 minutes, wants 95% confidence and a margin of error of 2 minutes:

The sample size is rounded upward to 139 deliveries. If the sampling design uses clusters or stratification, the calculation may need a design effect or allocation adjustment.

Confidence, precision and cost

Increasing confidence increases the critical value and therefore increases sample size. Reducing the margin of error has a much stronger effect because E is squared in the denominator. Sampling costs should be balanced against the value of more precise information. A sample-size calculation should also consider subgroup analysis: a total sample adequate for the whole population may be too small for reliable estimates within regions or customer segments.

8. Worked Interpretation of a Sampling Distribution

Suppose a population of transaction amounts has μ = $80 and σ = $24. A random sample of n = 64 transactions is selected. The sampling distribution of x̄ has:

Thus, sample means typically vary around $80, with a standard error of $3. If the population is sufficiently suitable for the CLT, a sample mean of $86 has a standardised value:

A value two standard errors above the population mean is relatively high but not impossible. The standard error describes the variability of sample means, not the variability of individual transaction amounts.

9. Applied Case: Customer Churn Survey

Background

Illustrative case: a subscription-based software company has 50,000 active business customers. Management wants to estimate the proportion considering cancellation within the next six months and compare results across small, medium and large customers.

Problem

A marketing employee proposes posting an open online survey link. The first 1,000 responses would be treated as a representative sample. This approach is inexpensive, but customers with strong opinions may be more likely to respond, and customers who do not use the company’s online community may be missed.

Analysis

The target population is all active business customers. The frame is the customer relationship management database, which contains customer size, region, contract value and contact details. Because customer size is central to the management decision, stratified probability sampling is appropriate. Each size category can be represented, and random selection can occur within each stratum.

Suppose the company seeks a 95% confidence estimate with a ±4 percentage-point margin of error. Using the conservative p* = 0.5 assumption:

At least 601 completed responses are needed for a simple random design. If management requires separate estimates for three customer-size groups, the total sample must be increased or allocated carefully. A proportional allocation may represent the population accurately, while a minimum allocation per group may support comparisons. If the response rate is expected to be 60%, approximately 601 ÷ 0.60 ≈ 1,002 customers should be invited, before allowing for ineligible or invalid contacts.

Approach and solution

The company selects customers randomly within each size stratum, sends a neutral invitation, provides a clear definition of “considering cancellation,” and follows up with nonrespondents. Response rates are monitored by stratum. Weighting may be required if final response patterns differ from population proportions. The survey instrument is tested in a pilot, and personally identifiable information is separated from analytical responses wherever possible.

Outcome

The resulting estimate is more defensible than the open-link result because selection is controlled and coverage is documented. Management can report the overall estimate and subgroup estimates with appropriate uncertainty. However, the company should still disclose nonresponse, frame limitations and the exact wording of the question.

Learning points

The case demonstrates that a credible sample involves more than obtaining a large number of responses. The target population, sampling frame, selection mechanism, response process, sample-size objective and analysis plan must work together.

10. Professional, Ethical and Technological Considerations

Sampling decisions can affect pricing, credit access, employee evaluation, public services and customer treatment. Analysts should avoid deliberately excluding inconvenient groups, hiding uncertainty or describing a volunteer sample as random. Consent, privacy, data minimisation and secure storage are essential when sampling involves personal information.

Automated systems and AI can help construct frames, detect duplicates, monitor response patterns and simulate sampling distributions. They do not eliminate the need for methodological judgement. An algorithm trained on an incomplete customer database may reproduce coverage bias, and automated invitations may disproportionately reach digitally active users. Analysts should document data sources, selection rules, exclusions, weighting procedures and quality checks.

Best practice includes:

• define the target population before selecting the sample;
• assess the completeness and currency of the sampling frame;
• use randomisation where generalisation is required;
• stratify when important subgroups need reliable representation;
• anticipate nonresponse and monitor it by subgroup;
• distinguish sampling error from bias and other errors;
• calculate sample size from a stated precision objective;
• report the method, response rate, weighting and limitations;
• preserve confidentiality and use data only for legitimate purposes.

Chapter Summary

Sampling allows analysts to learn about large populations from manageable subsets. Probability sampling methods—simple random, systematic, stratified, cluster and multistage sampling—use known selection probabilities and support statistical inference. Non-probability methods—including convenience, purposive, quota, snowball and volunteer sampling—can be useful for exploratory work but may not support defensible population margins of error.

Sampling error is random variation caused by observing a sample rather than a census. Nonsampling errors include coverage, nonresponse, measurement and processing errors; larger samples do not automatically remove them. A sampling distribution describes the behaviour of a statistic across repeated samples. The central limit theorem explains why sample means often have an approximately normal distribution for sufficiently large samples. The standard error of a mean is σ ÷ √n, while the standard error of a proportion is √[p(1 − p) ÷ n].

Sample-size planning links confidence, desired precision and variability. Proper planning also accounts for finite populations, nonresponse, subgroup analysis and complex designs. Professional sampling requires transparency, privacy protection, careful frame construction and honest communication of uncertainty.

Key Terms

Population: The complete set of units relevant to a study.

Target population: The population to which the researcher intends to generalise.

Sampling frame: The operational list from which a sample is selected.

Sample: A subset of the population observed in a study.

Parameter: A numerical characteristic of a population.

Statistic: A numerical characteristic calculated from a sample.

Probability sampling: Sampling in which each unit has a known, non-zero selection probability.

Non-probability sampling: Sampling in which selection probabilities are unknown.

Stratified sampling: Sampling separately from defined population subgroups.

Cluster sampling: Sampling naturally occurring groups and studying units within them.

Systematic sampling: Selecting units at a regular interval after a random start.

Sampling error: Random difference between a sample statistic and population parameter.

Nonsampling error: Error arising from coverage, nonresponse, measurement or processing.

Sampling distribution: Probability distribution of a statistic over repeated samples.

Standard error: Standard deviation of a sampling distribution.

Central limit theorem: Result stating that sample means tend toward an approximately normal distribution under broad conditions.

Margin of error: Maximum planned distance between an estimate and the parameter at a stated confidence level.

Finite population correction: Adjustment to sampling variability when a large fraction of a finite population is sampled.

Design effect: Inflation of variance caused by a complex sampling design relative to simple random sampling.

Review Questions

Short Answer Questions

1. Distinguish a population parameter from a sample statistic.
2. What is the main difference between probability and non-probability sampling?
3. Define sampling error and explain why it cannot be eliminated completely by selecting a random sample.
4. What does the standard error of a sample mean measure?
5. State two conditions that support use of the central limit theorem.

Long Answer Questions

1. Compare simple random, systematic, stratified, cluster and multistage sampling. Give one suitable business application for each.
2. Explain how coverage error, nonresponse error and measurement error can affect a customer survey.
3. Derive and interpret the relationship between sample size and the standard error of the sample mean.
4. Explain the purpose of a sampling distribution and distinguish it from the distribution of individual observations.
5. Discuss the ethical responsibilities of an analyst reporting results from a non-probability sample.

Analytical / Application Questions

1. A retailer has 20,000 loyalty-card customers and wants a 95% confidence estimate of the proportion who prefer home delivery, with a margin of error of ±5 percentage points. Assuming no prior estimate of the proportion, calculate the initial sample size and explain whether a finite-population correction is relevant.
2. A survey of employee satisfaction receives 2,000 voluntary responses from a workforce of 15,000. The analyst reports a margin of error of ±2%. Evaluate this claim and recommend a more defensible reporting approach.
3. A company samples 400 invoices using a systematic interval of 25. Identify the implied frame size, explain why the starting point should be random, and describe a potential periodicity problem.

Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes a parameter from a statistic?

A. A parameter describes a sample, while a statistic describes a population.

B. A parameter describes a population, while a statistic is calculated from a sample.

C. A parameter is always qualitative, while a statistic is always quantitative.

D. A parameter is selected randomly, while a statistic is selected systematically.

Correct Answer: B. A parameter describes a population, while a statistic is calculated from a sample.

Question 2

A researcher divides customers into regions and randomly selects customers from every region. Which method is being used?

A. Cluster sampling

B. Convenience sampling

C. Stratified sampling

D. Snowball sampling

Correct Answer: C. Stratified sampling

Question 3

If a population standard deviation is 20 and the sample size is 100, what is the standard error of the sample mean?

A. 0.2

B. 2

C. 10

D. 20

Correct Answer: B. 2

Question 4

Which error is most directly caused by selected people failing to respond?

A. Coverage error

B. Nonresponse error

C. Processing error

D. Sampling-frame duplication

Correct Answer: B. Nonresponse error

Question 5

What does the central limit theorem primarily describe?

A. The distribution of individual observations becoming uniform

B. The sampling distribution of sample means becoming approximately normal under suitable conditions

C. The population mean becoming equal to zero

D. The sample standard deviation becoming equal to the population mean

Correct Answer: B. The sampling distribution of sample means becoming approximately normal under suitable conditions

Question 6

With no prior estimate of a population proportion, which planning value is conventionally conservative?

A. p* = 0.01

B. p* = 0.10

C. p* = 0.50

D. p* = 1.00

Correct Answer: C. p* = 0.50

Question 7

A sample size is quadrupled while all other conditions remain unchanged. Approximately what happens to the standard error of a sample mean?

A. It doubles

B. It is cut in half

C. It is reduced to one quarter

D. It does not change

Correct Answer: B. It is cut in half

Question 8

Which sampling method selects every kth unit after a random starting point?

A. Systematic sampling

B. Purposive sampling

C. Cluster sampling

D. Quota sampling

Correct Answer: A. Systematic sampling

Question 9

A volunteer online poll obtains 10,000 responses. Which conclusion is most appropriate?

A. The large sample guarantees representativeness.

B. The result has no possible bias.

C. The result may still suffer from self-selection bias.

D. The result is automatically a census.

Correct Answer: C. The result may still suffer from self-selection bias.

Question 10

If N = 12,000 and the desired systematic sample size is n = 400, what is the approximate sampling interval?

A. 3

B. 12

C. 30

D. 400

Correct Answer: C. 30