Unit 4: Sampling | Research Methodology

Unit 4 Contents

4.1 Introduction Basic Sampling Terms Sampling Steps 4.2 Types of Sampling Probability Sampling Non-Probability Sampling Sampling Errors 4.3 Sample Size Numerical Examples Reliability of Sample Quick Revision Exam Preparation

Unit Overview

Sampling is an important part of research methodology. In many research studies, it is difficult or impossible to study every individual, object or observation in the entire population. Therefore, a smaller group called a sample is selected from the population and studied.

The quality of the sample directly affects the quality and generalizability of research findings. A properly selected sample should represent the population as closely as possible and should be selected using an appropriate sampling method.

Sampling: Sampling is the process of selecting a subset of individuals, objects or observations from a defined population for the purpose of conducting research.

Exam Focus: Prepare definitions of population, sample, sampling unit, sampling frame, sampling error and non-sampling error. Also prepare probability and non-probability sampling methods and sample-size calculations.

Population Sample Sampling Frame Probability Non-Probability Sample Size Sampling Error

4.1 Introduction and Review of Sampling

Need for Sampling

Sampling is needed because studying the entire population may require excessive time, money, manpower and resources. In some studies, examination of every member may also destroy or consume the material being tested.

Major Reasons for Sampling

Large population size.
Limited time available for research.
Limited financial resources.
Limited manpower and equipment.
Need for faster data collection and analysis.
Destructive nature of some tests.
Difficulty in accessing the entire population.
Need for manageable research work.

Advantages of Sampling

Less expensive than a census.
Requires less time.
Requires fewer investigators.
Allows detailed investigation.
May provide better control over data collection.
Useful when the population is very large.

Disadvantages of Sampling

Sampling error may occur.
An inappropriate sample may not represent the population.
Sampling requires careful planning.
Some population groups may be missed.
Results may be less precise than a complete census.

Basic Terms in Sampling

1. Population

Population is the complete set of individuals, objects or observations about which the researcher wants to draw conclusions.

Example: If a researcher wants to study antibiotic resistance among all bacterial isolates obtained from a particular hospital during one year, all eligible isolates constitute the population.

2. Sample

Sample is a selected subset of the population that is actually included in the research.

If 100 bacterial isolates are selected from a population of 1,000 isolates, the 100 isolates form the sample.

3. Sampling Unit

A sampling unit is the individual element or group of elements that can be selected during sampling.

Examples: A student, household, patient, water sample, bacterial isolate, village or hospital may serve as a sampling unit depending on the study.

4. Sampling Frame

Sampling frame is a list or other practical representation of all sampling units from which the sample is selected.

For a study of students, the official list of eligible students may serve as the sampling frame.

5. Sampling Error

Sampling error is the difference between a population value and the value estimated from a sample due to the fact that only part of the population has been studied.

Sampling Error = Sample Estimate − Population Parameter

The sign may be positive or negative depending on whether the sample estimate is higher or lower than the actual population value.

6. Non-Sampling Error

Non-sampling error is an error that does not arise simply because a sample is used. It can occur during measurement, data collection, recording, processing, non-response or other stages of research.

Sampling Error	Non-Sampling Error
Occurs because a sample rather than the whole population is studied.	Can occur even in a census.
Related to sample selection and sample-to-population differences.	May result from poor measurement, recording, non-response or data processing.
Can often be reduced by improving sample design and increasing sample size.	Requires quality control, training and careful data collection.

Steps in Sampling

Step 1: Define the target population
Clearly identify the population to which the research findings will apply.

Step 2: Define the sampling unit
Determine the basic unit that can be selected.

Step 3: Prepare or identify the sampling frame
Develop a complete and appropriate list of eligible sampling units where possible.

Step 4: Select the sampling method
Choose an appropriate probability or non-probability sampling method.

Step 5: Determine sample size
Decide how many units should be included.

Step 6: Select the sample
Apply the selected sampling procedure.

Step 7: Collect data
Collect information from the selected sample.

Step 8: Assess sample quality
Check representativeness, completeness and possible sources of error.

Characteristics of a Good Sample

Representative of the target population.
Adequate in size.
Selected using an appropriate method.
Free from avoidable selection bias.
Based on clearly defined eligibility criteria.
Suitable for the objectives of the research.
Feasible within available resources.
Provides sufficiently reliable information.

4.2 Types of Sampling

Sampling methods are broadly divided into two major categories:

Probability Sampling	Non-Probability Sampling
Every eligible sampling unit has a known or specified chance of selection.	Selection probability is not necessarily known.
Uses randomization.	Usually does not require random selection.
Generally supports statistical inference to the population when assumptions are met.	Generalization may be more limited and depends on study design.
Examples: simple random, stratified, systematic, cluster, multistage.	Examples: convenience, purposive, quota, snowball, self-selecting.

Probability Sampling

1. Simple Random Sampling

Simple random sampling is a method in which every eligible sampling unit has a known and generally equal chance of being selected.

Methods of Selection

Lottery method
Random number table
Computer-generated random numbers

Example: A researcher has a list of 500 students and needs 50 students. Each student is assigned a number from 1 to 500, and 50 numbers are selected randomly.

Advantages

Simple concept.
Reduces personal selection bias.
Easy to analyze statistically.

Disadvantages

Requires a suitable sampling frame.
May be difficult for widely scattered populations.
Small subgroups may be underrepresented by chance.

2. Stratified Sampling

Stratified sampling involves dividing the population into relatively homogeneous subgroups called strata and selecting samples from each stratum.

Example: A college population may be divided into first-year, second-year and third-year students, followed by random selection within each group.

Proportionate Stratified Sampling

The number selected from each stratum is proportional to the size of that stratum.

nₕ = (Nₕ / N) × n

Where:

nₕ = sample size from stratum h
Nₕ = population size of stratum h
N = total population size
n = total sample size

Numerical Example 1: Proportionate Stratified Sampling

A college has 1,000 students:

Science = 400
Management = 350
Humanities = 250

A sample of 100 students is required. Find the number selected from each stratum using proportionate stratified sampling.

Science:
nₕ = (400 / 1000) × 100
= 40 students

Management:
nₕ = (350 / 1000) × 100
= 35 students

Humanities:
nₕ = (250 / 1000) × 100
= 25 students

Answer: Science = 40, Management = 35, Humanities = 25.
Total = 100 students.

3. Systematic Sampling

Systematic sampling selects units from an ordered sampling frame at a regular interval after a random starting point.

Sampling Interval

k = N / n

Where: N = population size and n = required sample size.

Numerical Example 2: Systematic Sampling

A population contains 1,000 households and a sample of 100 households is required. Find the sampling interval.

Given:
N = 1000
n = 100

Formula:
k = N / n

k = 1000 / 100
k = 10

Therefore, every 10th household is selected after choosing an appropriate random starting point between 1 and 10.

Answer: Sampling interval = 10.

4. Cluster Sampling

Cluster sampling divides the population into naturally occurring groups called clusters and selects clusters for study.

Example: To study school students across a large district, schools may be treated as clusters. A number of schools are selected and students within selected schools are studied.

Advantages

Useful for geographically dispersed populations.
Can reduce travel and administrative costs.
Practical for large populations.

Disadvantages

Selected clusters may differ from each other.
Sampling error may be greater than simple random sampling under some designs.

5. Multistage Sampling

Multistage sampling involves selecting the sample through two or more stages.

Example:
Country → Province → District → Municipality → Household → Individual

Different sampling methods can be combined at different stages.

Remember: Probability sampling methods included in the syllabus: Simple random, stratified, systematic, cluster and multistage sampling.

Process of Selecting a Random Sample

Define the population.
Prepare a sampling frame.
Assign a unique number to each unit.
Determine the required sample size.
Generate random numbers.
Select units corresponding to those numbers.
Check eligibility and completeness.

Numerical Example 3: Selecting Random Numbers

Suppose 20 students are numbered 1–20 and five students are required. A random selection gives:

3, 7, 11, 16, 19

The selected sample consists of students numbered 3, 7, 11, 16 and 19.

Non-Probability Sampling

1. Convenience Sampling

Convenience sampling selects participants or units that are readily available and easy to access.

A researcher surveys students who are present in the laboratory at the time of data collection.

Advantage: Fast, inexpensive and easy.
Disadvantage: May have poor representativeness and selection bias.

2. Purposive Sampling

Purposive sampling selects units deliberately because they possess particular characteristics relevant to the research.

A researcher studying antimicrobial resistance may deliberately select laboratories with relevant bacterial culture facilities.

Advantage: Useful when specialized participants are required.
Disadvantage: Selection depends on researcher judgment.

3. Quota Sampling

Quota sampling divides the population into categories and selects a specified number of units from each category without necessarily using random selection.

A researcher may decide to interview 50 males and 50 females and continue recruiting until both quotas are filled.

4. Snowball Sampling

Snowball sampling begins with a small number of eligible participants who identify or recruit other potential participants.

It is particularly useful for populations that are difficult to identify or access through conventional sampling frames.

5. Self-Selecting Sampling

Self-selecting sampling occurs when individuals volunteer themselves for participation after being invited or given an opportunity to participate.

An online survey allows anyone who sees the survey link to volunteer to respond.

Comparison of Non-Probability Sampling Methods

Method	Basis of Selection	Common Use	Main Limitation
Convenience	Easy availability	Quick exploratory studies	High risk of selection bias
Purposive	Researcher's judgment	Specialized populations	Subjective selection
Quota	Predefined category quotas	Ensuring category numbers	Within-quota selection may be biased
Snowball	Participant referrals	Hard-to-reach populations	Network-dependent selection
Self-selecting	Voluntary participation	Open surveys and invitations	Volunteer bias

Sampling Error and Non-Sampling Error

Sampling Error

Sampling error arises because the sample contains only a portion of the population. Different samples from the same population may produce slightly different estimates.

Numerical Example 4: Sampling Error

Suppose the actual population mean is 60 and the sample mean is 57. Calculate the sampling error.

Sampling Error = Sample Estimate − Population Parameter

= 57 − 60

= −3

Answer: Sampling error = −3.
The sample mean is 3 units lower than the population mean.

Absolute Sampling Error

Absolute Sampling Error = |Sample Estimate − Population Parameter|

Numerical Example 5: Absolute Sampling Error

Population proportion = 0.60
Sample proportion = 0.55

Absolute error = |0.55 − 0.60|

= |−0.05|

Absolute sampling error = 0.05

Non-Sampling Errors

Measurement error
Interviewer error
Recording error
Data-entry error
Non-response error
Processing error
Coverage error
Questionnaire-related error

4.3 Size of Sample

Sample size refers to the number of units included in a sample. Selecting an appropriate sample size is important because a sample that is too small may provide imprecise estimates, while an unnecessarily large sample may waste resources.

Factors Affecting Sample Size

Size of the population.
Variability of the population.
Required precision or margin of error.
Desired confidence level.
Study design.
Expected response rate.
Available time and resources.
Subgroup analysis requirements.
Expected effect size in analytical studies.
Statistical method planned for analysis.

Common Formula for Estimating Sample Size for a Proportion

For a large population when estimating a population proportion, a commonly used formula is:

n = Z²pq / e²

Where:

n = required sample size
Z = Z value corresponding to the desired confidence level
p = expected proportion
q = 1 − p
e = acceptable margin of error

Common Z Values

Confidence Level	Approximate Z Value
90%	1.645
95%	1.96
99%	2.576

Numerical Example 6: Sample Size for a Proportion

Calculate the required sample size when:

Confidence level = 95%
Expected proportion p = 0.50
Therefore q = 0.50
Margin of error e = 0.05

Formula:
n = Z²pq / e²

n = (1.96)² × 0.50 × 0.50 / (0.05)²

n = 3.8416 × 0.25 / 0.0025

n = 0.9604 / 0.0025

n = 384.16

Required sample size ≈ 385.
The result is rounded upward to the next whole number.

Why is p = 0.50 Often Used?

When there is no reliable prior estimate of the population proportion, p = 0.50 is commonly used because it gives the maximum value of pq and therefore produces a conservative sample-size estimate for this formula.

Numerical Example 7: Sample Size with p = 0.20

Find the sample size for a 95% confidence level, expected proportion of 20% and margin of error of 5%.

p = 0.20
q = 1 − 0.20 = 0.80
Z = 1.96
e = 0.05

n = (1.96)² × 0.20 × 0.80 / (0.05)²

n = 3.8416 × 0.16 / 0.0025

n = 0.614656 / 0.0025

n = 245.8624

Required sample size ≈ 246.

Finite Population Correction

When the population is relatively small and a substantial fraction of it is being sampled, a finite population correction can be applied.

n_f = n / [1 + (n − 1) / N]

Where:

n_f = corrected sample size
n = sample size calculated for a large population
N = total population size

Numerical Example 8: Finite Population Correction

Suppose the initial sample size is 385 and the total population is 1,000. Calculate the corrected sample size.

n_f = n / [1 + (n − 1) / N]

= 385 / [1 + (385 − 1) / 1000]

= 385 / [1 + 384/1000]

= 385 / 1.384

≈ 278.18

Corrected sample size ≈ 279.

Sample Size Adjustment for Non-Response

If some selected participants are expected not to respond, the initial sample can be increased so that the final number of completed observations is adequate.

Adjusted Sample Size = Required Completed Sample / Expected Response Rate

Numerical Example 9: Adjustment for Non-Response

A study requires 385 completed responses. The expected response rate is 80%. How many participants should be approached?

Response rate = 80% = 0.80

Adjusted sample size = 385 / 0.80

= 481.25

Approximately 482 participants should be approached.

Sampling Fraction

Sampling fraction is the proportion of the population included in the sample.

Sampling Fraction = n / N

It may also be expressed as a percentage:

Sampling Fraction (%) = (n / N) × 100

Numerical Example 10: Sampling Fraction

A population contains 2,000 individuals and 200 are selected. Calculate the sampling fraction.

Sampling fraction = 200 / 2000

= 0.10

= 0.10 × 100

Sampling fraction = 10%

Sample Size for Estimating a Mean

When the objective is to estimate a population mean and an estimate of the population standard deviation is available, a commonly used formula is:

n = (Zσ / E)²

Where:

Z = Z value for desired confidence level
σ = estimated population standard deviation
E = acceptable margin of error

Numerical Example 11: Sample Size for a Mean

Suppose the estimated population standard deviation is 10 units, the desired confidence level is 95%, and the acceptable margin of error is 2 units. Find the sample size.

Z = 1.96
σ = 10
E = 2

n = (Zσ / E)²

n = (1.96 × 10 / 2)²

n = (9.8)²

n = 96.04

Required sample size ≈ 97.

Testing the Reliability of a Sample

A sample is considered useful when it provides sufficiently reliable information about the target population. Reliability of a sample is influenced by the sampling method, sample size, population variability, non-response and selection bias.

Methods for Improving Sample Reliability

Use an appropriate sampling frame.
Use random selection when probability sampling is appropriate.
Use adequate sample size.
Use clearly defined inclusion and exclusion criteria.
Reduce non-response.
Maintain accurate records.
Use appropriate data collection instruments.
Monitor sampling procedures.

Relationship Between Sample Size and Precision

In general, increasing sample size reduces random sampling variability and improves the precision of estimates, although increasing sample size does not automatically eliminate systematic bias or non-sampling errors.

Important: A very large biased sample can still produce misleading results. Good sampling requires both an appropriate sampling method and an adequate sample size.

Process of Selecting a Random Sample

1. Identify population: Define exactly who or what belongs to the study population.

2. Prepare sampling frame: Prepare the list of eligible units.

3. Determine sample size: Calculate or decide the required number of units.

4. Assign identification numbers: Give each unit a unique number.

5. Generate random numbers: Use a lottery method, random-number table or computer.

6. Select units: Choose the units corresponding to the selected random numbers.

7. Verify eligibility: Ensure that selected units satisfy the inclusion criteria.

Additional Numerical Practice

Numerical 12: Proportionate Stratified Sampling

A population contains 2,000 people:

Group A = 800
Group B = 600
Group C = 400
Group D = 200

If the total sample is 200, calculate the sample from each group.

Show Solution

Group A:

(800 / 2000) × 200 = 80

Group B:

(600 / 2000) × 200 = 60

Group C:

(400 / 2000) × 200 = 40

Group D:

(200 / 2000) × 200 = 20

Answer: A = 80, B = 60, C = 40, D = 20.
Total = 200.

Numerical 13: Systematic Sampling

A list contains 1,500 households and the researcher needs 150 households. Find the sampling interval.

Show Solution

k = N / n

= 1500 / 150

= 10

Sampling interval = 10.

Numerical 14: Sample Size for Proportion

Calculate sample size for 95% confidence, p = 0.40 and margin of error = 0.05.

Show Solution

q = 1 − 0.40 = 0.60

n = Z²pq / e²

= (1.96)² × 0.40 × 0.60 / (0.05)²

= 3.8416 × 0.24 / 0.0025

= 0.921984 / 0.0025

= 368.79

Required sample size ≈ 369.

Numerical 15: Sampling Fraction

A researcher selects 75 samples from a population of 500. Calculate the sampling fraction as a percentage.

Show Solution

Sampling Fraction (%) = (n / N) × 100

= (75 / 500) × 100

= 15%

Sampling fraction = 15%.

Numerical 16: Non-Response Adjustment

A researcher requires 300 completed questionnaires and expects an 85% response rate. Calculate the number of people who should be approached.

Show Solution

Adjusted sample = Required completed sample / Response rate

= 300 / 0.85

= 352.94

Approximately 353 participants should be approached.

Numerical 17: Sampling Error

The actual population proportion is 0.72 and the sample proportion is 0.68. Calculate the sampling error and absolute sampling error.

Show Solution

Sampling Error = 0.68 − 0.72

= −0.04

Absolute Sampling Error = |−0.04| = 0.04

Sampling error = −0.04
Absolute sampling error = 0.04

Probability Sampling: Quick Comparison

Method	Main Principle	Best Feature	Limitation
Simple Random	Random selection of individuals	Simple and reduces selection bias	Needs sampling frame
Stratified	Divide into strata and sample from each	Can ensure subgroup representation	Requires information about strata
Systematic	Select every kth unit	Easy to conduct	Can be affected by periodic patterns
Cluster	Select natural groups	Efficient for dispersed populations	Clusters may differ substantially
Multistage	Sampling in multiple stages	Practical for large populations	More complex design

Important Differences

Population vs Sample

Population	Sample
Entire group of interest.	Subset selected from the population.
Usually larger.	Usually smaller.
Population parameter describes it.	Sample statistic describes it.
May be difficult or expensive to study completely.	Usually more practical to study.

Probability vs Non-Probability Sampling

Probability Sampling	Non-Probability Sampling
Uses random selection.	Does not necessarily use random selection.
Selection probability is known or specified.	Selection probability is not necessarily known.
Generally better for statistical generalization when assumptions are met.	Generalization is more limited.
Usually requires more planning.	Often easier and faster.
Examples: simple random, stratified, systematic.	Examples: convenience, purposive, quota.

Sampling Formula Sheet

1. Sampling Interval
k = N / n

2. Proportionate Stratified Sample
nₕ = (Nₕ / N) × n

3. Sampling Error
Sampling Error = Sample Estimate − Population Parameter

4. Absolute Sampling Error
|Sample Estimate − Population Parameter|

5. Sample Size for Proportion
n = Z²pq / e²

6. q
q = 1 − p

7. Finite Population Correction
n_f = n / [1 + (n − 1)/N]

8. Sample Size Adjustment for Non-response
Adjusted Sample = Required Completed Sample / Response Rate

9. Sampling Fraction
Sampling Fraction = n / N

10. Sampling Fraction Percentage
Sampling Fraction (%) = (n / N) × 100

11. Sample Size for Mean
n = (Zσ / E)²

Quick Revision

Term	Quick Meaning
Population	Entire group of interest.
Sample	Selected subset of population.
Sampling Unit	Basic unit that can be selected.
Sampling Frame	List or representation from which sample is selected.
Sampling Error	Difference caused by studying a sample instead of the whole population.
Non-Sampling Error	Error arising from causes other than sampling itself.
Simple Random	Random selection of individual units.
Stratified	Divide population into strata and sample from each.
Systematic	Select every kth unit.
Cluster	Select natural groups or clusters.
Multistage	Select samples through multiple stages.
Convenience	Select easily accessible units.
Purposive	Select units based on relevant characteristics.
Quota	Fill predetermined category quotas.
Snowball	Participants help identify additional participants.
Self-selecting	Individuals volunteer themselves.

Examination Preparation

2-Mark Questions

Define sampling.
Define population.
What is a sample?
Define sampling unit.
What is a sampling frame?
Define sampling error.
What is non-sampling error?
State any two advantages of sampling.
Mention any two probability sampling methods.
Mention any two non-probability sampling methods.
Define simple random sampling.
What is stratified sampling?
What is systematic sampling?
Define cluster sampling.
What is multistage sampling?
Define convenience sampling.
What is purposive sampling?
Define quota sampling.
What is snowball sampling?
Define sampling fraction.

5-Mark Questions

Explain the need and importance of sampling.
Discuss the advantages and disadvantages of sampling.
Explain the steps involved in sampling.
Describe the characteristics of a good sample.
Explain simple random sampling with advantages and disadvantages.
Describe stratified sampling with an example.
Explain systematic sampling with a numerical example.
Describe cluster and multistage sampling.
Explain convenience, purposive and quota sampling.
Write short notes on snowball and self-selecting sampling.
Differentiate sampling error and non-sampling error.
Discuss factors affecting sample size.
Explain the methods of estimating sample size.

10-Mark Long Questions

Define sampling and discuss its importance. Explain population, sample, sampling unit, sampling frame, sampling error, non-sampling error and steps in sampling.
Discuss probability sampling methods in detail. Explain simple random, stratified, systematic, cluster and multistage sampling with suitable examples.
Discuss non-probability sampling methods. Explain convenience, purposive, quota, snowball and self-selecting sampling with advantages and disadvantages.
Discuss sample size determination. Explain factors affecting sample size and demonstrate appropriate numerical calculations.
Differentiate probability and non-probability sampling. Explain major methods under each category.
Explain sampling errors. Differentiate sampling and non-sampling errors with suitable examples.
Calculate the required sample size for a population proportion. Explain the formula and all terms involved.

Important Numerical Questions

Calculate the sampling interval using k = N/n.
Calculate proportionate sample size from different strata.
Calculate sampling error.
Calculate absolute sampling error.
Calculate sampling fraction.
Calculate sample size using n = Z²pq/e².
Calculate corrected sample size using finite population correction.
Adjust sample size according to expected response rate.
Calculate sample size for estimating a mean using n = (Zσ/E)².

High-Priority Topics

Must Prepare:

Population, sample, sampling unit and sampling frame
Sampling error vs non-sampling error
Steps in sampling
Simple random sampling
Stratified sampling + numerical
Systematic sampling + numerical
Cluster and multistage sampling
Non-probability sampling methods
Factors affecting sample size
Sample-size numerical using Z²pq/e²
Finite population correction
Non-response adjustment

Exam Writing Pattern for Sampling

For a 10-mark sampling question, use the following order:

Definition → Need → Basic terms → Steps → Classification → Detailed methods → Advantages/Disadvantages → Numerical example → Conclusion

For numerical questions, always write:

Given → Formula → Substitution → Calculation → Final Answer

Unit 4 Final Summary

Sampling is the process of selecting a manageable subset from a defined population for research. A good sampling design should provide an adequate and appropriate sample while minimizing avoidable selection bias and errors. Probability sampling includes simple random, stratified, systematic, cluster and multistage sampling, whereas non-probability sampling includes convenience, purposive, quota, snowball and self-selecting methods.

Sample size is influenced by population characteristics, desired precision, confidence level, variability, study design, resources and expected non-response. Numerical calculations commonly involve sampling intervals, proportionate allocation, sampling error, sampling fraction and sample-size formulas.

Final Memory Line:

Population → Sampling Frame → Sampling Method → Sample Size → Sample Selection → Data Collection → Analysis → Generalization