Sampling in One Minute
Sampling means studying a properly selected part of a population and using it to understand the whole population.
1. Meaning and Need for Sampling
A business may want to know the average life of all bulbs produced in a factory. Testing every bulb would be costly, slow and destructive. So, a smaller representative group is tested and the result is used to judge the full production.
Why do we use sampling?
- The population may be too large or infinite.
- A complete enquiry may require too much time and money.
- Testing may destroy the item, such as bulb-life or strength testing.
- A smaller enquiry can be supervised more carefully.
- Decisions may be needed quickly.
Three stages of a sampling problem
2. Four Basic Principles of a Sample Survey
Statistical Regularity
A fairly large sample selected at random tends, on average, to represent the population.
Inertia of Large Numbers
Other things remaining the same, larger samples generally give more stable and reliable results.
Optimisation
Select the design that gives the required accuracy at minimum cost, or maximum accuracy for the available cost.
Validity
The design must permit valid estimation and valid testing of population parameters.
3. Sampling or Census?
A census studies every unit of the population. A sample survey studies only selected units.
| Basis | Sample Survey | Census |
|---|---|---|
| Coverage | Selected units | Every unit |
| Time | Usually faster | Usually slower |
| Total cost | Usually lower | Usually higher |
| Sampling error | Present | Absent |
| Non-sampling error | Possible | Possible |
| Destructive testing | Suitable | Not possible |
| Information on every unit | Not available | Available |
Prefer Sampling When
- The population is very large.
- Testing is destructive.
- Time and cost are limited.
Prefer Census When
- The population is small.
- Every individual unit must be known.
- Missing one defect can be dangerous.
4. Errors in a Survey
Sampling Error
It arises because only a part of the population is studied.
- Defective sampling design
- Substitution of selected units
- Faulty definition of sampling units
- Wrong choice of statistic
- High variability in the population
Non-Sampling Error
It arises from collection, response, recording or measurement problems.
- Non-response or incomplete coverage
- Wrong measurement or recording
- Memory failure or incorrect answers
- Enumerator or respondent bias
- Communication gap
5. Essential Terms
The complete collection of all units under study.
The total number of units in the population.
A representative part selected from the population.
The number of units included in the sample.
The basic unit selected, such as a person, household, machine or product.
A complete and updated list of all sampling units.
Types of Population
| Type | Meaning | Example |
|---|---|---|
| Finite | Contains a countable number of units | Students in a college |
| Infinite | Contains unlimited or practically uncountable units | Future production of bulbs |
| Existent | Consists of real units | Workers in a factory |
| Hypothetical | Exists only conceptually | Outcomes from unlimited coin tosses |
6. Parameter, Statistic and Statistical Inference
| Concept | Meaning | Examples |
|---|---|---|
| Parameter | A numerical characteristic of the population | Population mean μ, population variance σ², population proportion P |
| Statistic | A numerical measure calculated from sample observations | Sample mean x̄, sample variance s², sample proportion p |
7. Sampling Distribution and Standard Error
Sampling Fluctuation
Different samples of the same size may contain different units. Therefore, the value of a statistic such as the sample mean may change from one sample to another. This variation is called sampling fluctuation.
Sampling Distribution
The probability distribution of a statistic obtained from all possible samples of a fixed size is called its sampling distribution.
Standard Error
The standard deviation of a statistic is called its standard error. It measures the precision of the statistic as an estimate of the population parameter.
Number of Possible Samples
8. Types of Sampling
Probability Sampling
Each population unit has a known chance of selection.
Simple Random · Stratified · MultistageMixed Sampling
Selection is partly random and partly based on a fixed rule.
Systematic SamplingNon-Probability Sampling
Selection depends on judgement rather than known probabilities.
Purposive or Judgement Sampling9. Probability Sampling Methods
Simple Random Sampling
Every unit has an equal chance of being selected. Selection may be with replacement or without replacement.
Stratified Sampling
The population is divided into internally homogeneous groups called strata. A sample is then selected from each stratum.
- Proportional or Bowley allocation: sample size from each stratum is proportional to its population size.
- Neyman allocation: allocation depends jointly on population size and standard deviation of the stratum.
Multistage Sampling
Selection is made through successive levels. For example: State → District → Village → Household.
10. Systematic and Purposive Sampling
Systematic Sampling
Select one starting unit at random and then choose every kth unit.
Sampling interval: k = N / n
Advantage: Simple, quick and inexpensive.
Risk: Hidden periodicity in the list may produce a biased sample.
Purposive Sampling
The sampler selects units according to personal judgement.
Advantage: Useful for specialised or exploratory enquiries.
Limitation: It is subjective and does not support valid statistical hypothesis testing.
Quick Method Selector
| Situation | Suitable Method |
|---|---|
| Population is fairly homogeneous | Simple Random Sampling |
| Population contains distinct groups | Stratified Sampling |
| Population is geographically widespread | Multistage Sampling |
| A complete ordered list is available | Systematic Sampling |
| Selection depends on expert judgement | Purposive Sampling |
11. Worked Examples
Example 1: Number of Samples
A population has 5 units and a sample of 3 units is drawn without replacement.
Example 2: With Replacement
A population has 3 units and samples of size 2 are drawn with replacement.
Example 3: Systematic Sample
A list contains 1,000 customers and a sample of 100 is required.
If the random start is 6, the selected units are 6, 16, 26, 36 and so on.
12. Sampling — Visual Revision Map
Read the complete chapter in one connected flow.