Welcome to Sampling Techniques!
In this chapter, we are going to explore how mathematicians and scientists collect data without having to ask every single person in the world a question. Think of it like tasting a spoonful of soup to see if the whole pot needs more salt—you don't need to eat the whole pot to find out! We will learn the "right" and "wrong" ways to pick that spoonful so that our results are fair and accurate.
Don't worry if this seems like a lot of definitions at first. Once you see how these techniques work in real life, they become much easier to remember!
1. The Basics: Population vs. Sample
Before we look at how to sample, we need to know what we are sampling from.
Key Terms
Population: The entire group of items or people that we are interested in. Example: All the students in your school, or every cod in the North Sea.
Sample: A smaller group selected from the population to be studied. Example: 30 students picked from your school to answer a survey.
Why Sample?
We use samples because studying an entire population is often:
1. Too expensive.
2. Too time-consuming.
3. Impossible (e.g., you can't test every lightbulb to see how long it lasts, because then you’d have no lightbulbs left to sell!).
Inference: The "Big Picture"
When we calculate the mean or variance of a sample, we are using those numbers to make an informal inference (an educated guess) about the whole population.
Quick Tip: Different samples will give different results. If you pick five students, their average height might be different from the next five students you pick. This is normal!
Key Takeaway
A population is the "whole pot," and the sample is the "spoonful." We use the sample to guess what the whole pot is like.
2. Random Sampling Techniques
In a random sample, every item in the population has a chance of being picked. This helps prevent bias (being unfair or leaning toward one result).
Simple Random Sampling (SRS)
This is the "Gold Standard." Every possible sample of the size you want has an equal chance of being selected.
How to do it: Assign every person in the population a number, then use a random number generator to pick your sample.
Analogy: Putting everyone’s name in a giant hat and shaking it well before picking names out.
Systematic Sampling
This is a more "orderly" way to sample.
How to do it:
1. Calculate your "step" size \( k \): \( k = \frac{\text{Population Size}}{\text{Sample Size}} \)
2. Pick a random starting point between 1 and \( k \).
3. Pick every \( k \)-th person after that.
Example: If you have 100 people and want a sample of 10, your step is 10. You might start at person 3, then pick person 13, 23, 33, and so on.
Stratified Sampling
Use this when your population has distinct groups (called strata) that might behave differently. For example, Year 12s and Year 13s.
How to do it: You take a random sample from each group, but the size of the sample is proportional to the size of the group in the population.
The Formula:
\( \text{Number to sample from group} = \frac{\text{Number in group}}{\text{Total population}} \times \text{Total sample size} \)
Key Takeaway
Random methods are generally fairer because they avoid human choice, which often leads to bias.
3. Non-Random Sampling Techniques
Sometimes, random sampling is too hard or expensive. In these cases, we use other methods, but we have to be careful about bias.
Opportunity Sampling
This is simply picking the people who are available at the time.
Example: Standing outside a gym and interviewing the first 20 people who walk out.
The Problem: It’s very biased! Those people might all have similar interests (like fitness) and won't represent the whole town.
Quota Sampling
This is like stratified sampling, but not random.
How to do it: An interviewer is told to find 20 men and 20 women to interview. They can pick anyone they want until they hit those "quotas."
Did you know? Street pollsters often use this. Once they have enough "men under 30," they will stop asking them and look specifically for other groups.
Cluster Sampling
The population is divided into groups called "clusters" (usually based on location).
How to do it: You pick a few clusters at random and then sample everyone inside those clusters.
Example: To survey UK schools, you might pick 5 cities at random and survey every student in just those 5 cities.
Self-Selected (Volunteer) Sampling
People choose to be part of the sample themselves.
Example: An online "Yes/No" poll on a news website.
The Problem: Only people with very strong opinions usually bother to respond, so the results are often extreme!
Key Takeaway
Non-random samples are easier to get but are much more likely to be biased and unrepresentative of the population.
4. Evaluating a Sampling Method
In your exam, you might be asked to critique a sampling method. Here is a Quick Review of what to look for:
1. Is there Bias?
Does the method exclude certain groups? (e.g., surveying people at 10:00 AM on a Tuesday excludes most people who work 9-to-5 jobs).
2. Is it Practical?
Do you have a list of the whole population (a sampling frame)? If not, you can't do a Simple Random Sample!
3. Is the Sample Size large enough?
Small samples are more likely to give "weird" results just by chance.
Common Mistakes to Avoid
Mistake: Thinking that "Random" means "haphazard."
Correction: Random in math has a strict meaning—using a random number generator or a lottery system. Picking people "randomly" on the street is actually Opportunity Sampling!
Memory Aid: The Sampling Mnemonic
To remember the types of sampling, try this: "Silly Snakes Study Quietly On Clouds."
Simple Random
Systematic
Stratified
Quota
Opportunity
Cluster
Summary: Chapter Takeaways
1. Population = Whole group; Sample = Part of the group.
2. Simple Random Sampling is fair but requires a full list of the population.
3. Stratified Sampling ensures all groups (strata) are represented proportionally.
4. Systematic Sampling uses a fixed interval (every \( k \)-th item).
5. Opportunity and Self-selected methods are common but often highly biased.
6. Always check if a sample is representative before trusting the results!