Statistical sampling

Welcome to Statistical Sampling!

Ever wondered how news channels predict election results before all the votes are counted? Or how scientists decide if a new medicine works without testing every single person on Earth? The secret is Statistical Sampling.

In this chapter, we are going to learn how to look at a small group of data to understand a much larger one. This is a core part of your Paper 2 exam, so let’s dive in!

1. Population vs. Sample: The Soup Analogy

Before we start picking data, we need to know who (or what) we are talking about. There are two big terms you need to master:

1. Population: This is the entire group that you are interested in. If you are studying the heights of students in your college, the population is every single student in that college.

2. Sample: This is a subset or a small part of the population that you actually collect data from. If you ask 50 students their heights, those 50 people are your sample.

The Soup Analogy

Imagine you are cooking a massive pot of vegetable soup.
- The Population is the entire pot of soup.
- The Sample is the single spoonful you taste to see if it needs more salt.

Quick Review: You don't need to eat the whole pot (the population) to know how it tastes. You just need one good spoonful (the sample) to make an inference (an educated guess) about the whole thing!

Key Takeaway: We use a sample to make informal inferences about a population because it is usually cheaper, faster, and easier than looking at everything.

2. Sampling Techniques

For your AQA AS Level, you specifically need to know about two ways of picking your "spoonful":

A. Simple Random Sampling

In a Simple Random Sample, every single member of the population has an equal chance of being selected. It’s like putting everyone’s name into a giant hat and pulling them out blindly.

How to do it:
1. Give every member of the population a unique number.
2. Use a random number generator (on your calculator or a computer) to pick the numbers you need.
3. The people/items assigned to those numbers become your sample.

The Pros: It is totally unbiased. No one is being picked because of where they are or who they know.
The Cons: You need a full list of the population (a "sampling frame"), which can be hard to get.

B. Opportunity Sampling

Opportunity Sampling (sometimes called convenience sampling) is exactly what it sounds like: taking the sample from people who are available at the time and who fit your criteria.

Example: Standing outside a supermarket at 10:00 AM on a Tuesday and asking the first 20 people who walk past for their opinion on a local issue.

The Pros: It is very easy, quick, and inexpensive.
The Cons: It is often biased. In the example above, you wouldn't get the opinions of people who are at work or school at 10:00 AM!

Memory Aid:
- Random = Fair (everyone has a ticket).
- Opportunity = Easy (grabbing whoever is nearby).

3. Critiquing Your Sample

In your exam, you might be asked to "critique" or "evaluate" a sampling method. This just means you need to explain why a sample might be "bad" or "good."

The biggest enemy in sampling is Bias. Bias happens when your sample doesn't actually represent the population.

Example: If you want to know the average pocket money of teenagers in the UK, but you only ask students at an expensive private school, your result will be biased. It won't represent the whole country.

Common Mistakes to Avoid:

• Don't ignore the "Who": If a question says a researcher asked their friends, it's Opportunity Sampling and likely biased.
• Small Sample Sizes: If your sample size \( n \) is too small (like asking only 2 people), it’s not very reliable.
• Different Samples, Different Results: Understand that if two different people take two different samples from the same population, they will likely get different conclusions. This is called sampling variation and is perfectly normal!

Key Takeaway: Always look for Bias. Ask yourself: "Does everyone in the population have a fair chance of being in this sample?"

4. Working with the Large Data Set (LDS)

For Paper 2, AQA expects you to be familiar with a Large Data Set (currently based on the "Family Food" report). While you don't need to memorize the numbers, you should know that:

• You can take samples from this large data set to investigate trends (like how much milk people buy in different regions).
• Using technology (like a spreadsheet or a calculator) makes it much easier to handle these large amounts of information.

Did you know? The Large Data Set contains real-world data from thousands of households! It’s not just made-up math problems; it's how the government actually tracks what people eat.

Summary Checklist

Don't worry if this seems like a lot of definitions! Just remember these five things:

1. Population is the whole; Sample is the part.
2. Simple Random Sampling gives everyone an equal chance (use a random number generator!).
3. Opportunity Sampling is quick and easy but often biased.
4. A sample is used to make an inference about the population.
5. Different samples from the same population will give different results.

Success Tip: In exam questions, if you're asked why a method is bad, use the word "representative." For example: "The sample is not representative of the population because..."