Introduction to Statistical Sampling
Welcome to Statistical Sampling! Whenever researchers, businesses, or governments want to find out information—like who will win the next election, whether a batch of light bulbs is faulty, or what television shows teenagers prefer—they cannot always ask every single person or test every single product. Instead, they collect data from a smaller group.
In this chapter, you will learn how to select samples fairly, how to evaluate different sampling methods, and how to classify different types of statistical data. Don't worry if statistical terminology feels new; we will break down each idea step by step with relatable everyday examples!
1. Key Statistical Vocabulary: Populations and Samples
Before looking at how to take a sample, let us define the core terms used across all statistics exams.
Population vs. Sample
Population: The entire collection of all individuals, items, or units of interest to an investigator.
Example: All students currently enrolled at a school in Northern Ireland.
Census: An investigation or survey where information is collected from every single member of the entire population.
Real-world connection: The UK national census, conducted every 10 years, attempts to count every household in the country.
Sample: A smaller selection of individuals or items taken from the population to represent the whole group.
Analogy: Imagine tasting a single spoonful of soup to check the seasoning. The entire pot of soup is the population, and the spoonful is the sample. You don't need to drink the entire pot to know if it needs more salt!
Census vs. Sample: Pros and Cons
Advantages of a Census:
• It is completely accurate and provides a full picture.
• There is no sampling bias since every member is included.
Disadvantages of a Census:
• It is very time-consuming and expensive to carry out.
• It generates a massive amount of data that is difficult to process.
• It cannot be used if testing involves destructive testing (e.g., testing the lifespan of a battery or the safety of a car crash test—if you tested the whole population, you would destroy all products!).
Advantages of a Sample:
• Much quicker and cheaper than a census.
• Easier to organize and manage.
• Requires fewer resources and fewer people to process the data.
Disadvantages of a Sample:
• The sample may not be fully representative of the whole population.
• Introduces sampling error (natural differences between the sample and the true population).
Sampling Units and the Sampling Frame
Sampling Unit: An individual member or item of the population that can be sampled (e.g., one student, one household, or one computer chip).
Sampling Frame: A complete, uniquely identifiable list or register of all the sampling units in the population.
Example: The electoral register (list of voters), a school register of all students, or an inventory list of parts in a factory.
Important Exam Note: A sampling frame is an actual written list or database, not just a concept.
Key Takeaway:
A census observes everyone, while a sample observes a subset. To take a formal sample, you generally need a numbered list of all members, which is called the sampling frame.
2. Random (Probability) Sampling Methods
In a random sample (or probability sample), every item in the population has a known, non-zero probability of being selected. This eliminates human bias in selection.
Method 1: Simple Random Sampling
In a simple random sample of size \(n\), every individual item has an equal chance of being chosen, and every possible sample of size \(n\) has an equal chance of being selected.
Step-by-Step Process:
1. Allocate a unique number from \(1\) to \(N\) to each member of the sampling frame (where \(N\) is the total population size).
2. Generate \(n\) distinct random numbers using a random number table, a calculator (e.g., the Ran# or RanInt function), or a computer generator.
3. Match the selected random numbers to the corresponding items on the sampling frame.
4. Ignore and replace any duplicate numbers generated.
Advantages:
• Completely free from researcher bias.
• Simple to understand and easy to apply for small populations.
• Each item has an equal chance of selection.
Disadvantages:
• A complete sampling frame is strictly required.
• Not suitable for very large populations spread over a wide geographical area (can be costly and impractical).
• Can accidentally produce an unrepresentative sample by chance (e.g., picking only boys from a mixed-gender school).
Method 2: Systematic Sampling
In systematic sampling, items are chosen at regular, fixed intervals from an ordered sampling frame.
Step-by-Step Process:
1. Number each item in the sampling frame from \(1\) to \(N\).
2. Calculate the sampling interval: \(k = \frac{N}{n}\), where \(N\) is population size and \(n\) is desired sample size (round down to the nearest integer if needed).
3. Select a random starting point between \(1\) and \(k\). (Let's call this number \(r\)).
4. Select the members at positions: \(r, r + k, r + 2k, r + 3k, \dots\) until the full sample of size \(n\) is collected.
Example: To choose a sample of \(n = 20\) from a school of \(N = 600\) pupils:
• Interval: \(k = \frac{600}{20} = 30\).
• Pick a random number between \(1\) and \(30\). Suppose we pick \(12\).
• The sample will consist of pupil numbers: \(12, 42, 72, 102, \dots, 582\).
Advantages:
• Simple and quick to execute.
• Can be used for quality control on production lines (e.g., checking every \(50\)th item off a conveyor belt).
Disadvantages:
• A sampling frame is required.
• If the population list has a hidden cyclic or periodic pattern matching the interval \(k\), the sample can become heavily biased.
Method 3: Stratified Sampling
In stratified sampling, the population is divided into mutually exclusive, non-overlapping groups called strata (such as age brackets, genders, or year groups). A simple random sample is then taken from each stratum in proportion to the stratum's size in the population.
Stratified Sampling Formula:
\(\text{Number sampled from a stratum} = \frac{\text{Number in stratum}}{\text{Total population}} \times \text{Total sample size}\)
In symbols: \(\text{Sample size for stratum } i = \frac{N_i}{N} \times n\)
Worked Example: A college has \(800\) students: \(480\) in Year 13 and \(320\) in Year 14. A stratified sample of \(50\) students is to be selected.
• Year 13 sample size: \(\frac{480}{800} \times 50 = 30\)
• Year 14 sample size: \(\frac{320}{800} \times 50 = 20\)
• Check: \(30 + 20 = 50\) students in total.
Advantages:
• Guarantees that all sub-groups (strata) are represented proportionally.
• Reduces sampling error and gives more precise estimates for key sub-groups.
Disadvantages:
• Requires exact knowledge of population structure and a separate sampling frame for each stratum.
• Can be time-consuming to divide the population into distinct strata.
Key Takeaway:
Simple Random: Lottery style. Systematic: Every \(k\)th item. Stratified: Proportional sub-groups.
3. Non-Random (Non-Probability) Sampling Methods
In non-random sampling, items do not have a known or equal chance of being selected. These methods are common in market research because they are fast and do not require a sampling frame.
Method 1: Quota Sampling
In quota sampling, the researcher divides the population into groups based on specific characteristics (e.g., age, gender) and sets a target number ("quota") for each group. The researcher then interviews anyone who fits the criteria until the quota for each group is filled.
Example: A high street interviewer is told to survey \(25\) women over 50, \(25\) women under 50, \(25\) men over 50, and \(25\) men under 50. Once a category is full, they ignore anyone else in that category.
Stratified vs. Quota (A Common Exam Question!):
• Stratified Sampling: Members from each stratum are chosen randomly using a sampling frame.
• Quota Sampling: Members are chosen by the interviewer using non-random selection (no sampling frame needed).
Advantages:
• No sampling frame is needed.
• Quick, cost-effective, and easy to administer.
• Ensures key characteristics of the population are present in the sample.
Disadvantages:
• Non-random: introduces interviewer bias (e.g., choosing approachable people).
• Refusals can distort results (e.g., busy people decline to answer).
Method 2: Opportunity (Convenience) Sampling
In opportunity sampling, the sample is chosen from members of the population who are readily available and easiest to access at that time and place.
Example: Standing outside a gym at 8:00 AM and asking the first \(30\) people who walk past about their diet.
Advantages:
• Extremely easy, inexpensive, and fast to conduct.
• No sampling frame required.
Disadvantages:
• Highly unrepresentative and prone to bias (e.g., people outside a gym at 8:00 AM do not represent the general population's fitness habits!).
Key Takeaway:
Non-random methods (Quota and Opportunity) do not require a list (sampling frame), but they are prone to selection bias.
4. Types of Statistical Data
Once you collect a sample, you need to understand what type of data you have recorded.
Qualitative vs. Quantitative
• Qualitative Data (Categorical): Non-numerical observations (e.g., hair colour, car make, favourite subject).
• Quantitative Data (Numerical): Numerical measurements or counts (e.g., shoe size, mass, time, temperature).
Discrete vs. Continuous Quantitative Data
• Discrete Data: Can only take specific, distinct, separate values (often obtained by counting).
Examples: Number of siblings (\(0, 1, 2, 3\)), shoe sizes (\(6, 6.5, 7\)), score on a standard die (\(1, 2, 3, 4, 5, 6\)).
• Continuous Data: Can take any numerical value within a given interval or range (obtained by measuring).
Examples: Height (\(1.75\text{ m}\), \(1.754\text{ m}\)), weight, time, temperature.
Memory Trick: If you count it, it's discrete. If you measure it with a tool (like a ruler or stopwatch), it's continuous.
Grouped Data: Class Boundaries, Midpoints, and Widths
Continuous data (or large sets of discrete data) are often presented in groups known as classes.
Consider the continuous interval: \(20 \le x < 30\)
• Lower Class Boundary: \(20\)
• Upper Class Boundary: \(30\)
• Class Width: \(\text{Upper Boundary} - \text{Lower Boundary} = 30 - 20 = 10\)
• Midpoint: \(\frac{\text{Lower Boundary} + \text{Upper Boundary}}{2} = \frac{20 + 30}{2} = 25\)
Handling Gaps in Rounded Data:
Suppose age is recorded in completed whole years as \(20 - 29\), followed by \(30 - 39\).
Because data transitions at \(29.5\), the true boundaries for the \(20 - 29\) group are:
• Lower boundary: \(19.5\)
• Upper boundary: \(29.5\)
• Class width: \(29.5 - 19.5 = 10\)
• Midpoint: \(\frac{19.5 + 29.5}{2} = 24.5\)
5. Common Pitfalls and Exam Success Tips
1. Forgetting the "Random Start" in Systematic Sampling:
When describing systematic sampling, students often forget to state that the first item between \(1\) and \(k\) must be chosen at random.
2. Confusing Quota and Stratified Sampling:
Always check whether selection is random or not. If an interviewer picks people on the street to meet targets, it is quota sampling. If individuals are chosen using a list of names and random numbers, it is stratified sampling.
3. Forgetting to Round Strata Sizes Appropriately:
When calculating sample sizes for strata, numbers must be rounded to whole individuals such that their sum equals the total target sample size \(n\).
4. Not Linking Criticisms to the Context:
When an exam question asks for a disadvantage of a sample method in a given scenario, refer specifically to the context (e.g., "Surveying shoppers on a Monday morning will exclude people who work standard hours" rather than just "it is biased").
Final Summary Checklist
• A census surveys the whole population; a sample surveys a subset.
• A sampling frame is the complete numbered list of sampling units.
• Random methods: Simple Random, Systematic, Stratified.
• Non-random methods: Quota, Opportunity.
• Data types: Qualitative (words/categories) vs. Quantitative (numbers). Quantitative is either Discrete (counted) or Continuous (measured).