Welcome to Statistical Sampling!
In this chapter, we are going to explore how mathematicians and scientists collect information about the world. Whether it's predicting the results of an election or testing if a new medicine works, we rarely have the time or money to ask every single person on the planet for their opinion. That is where sampling comes in!
By the end of these notes, you’ll understand the difference between a whole group and a small group, and you'll know exactly how to pick the right people to study. Don't worry if statistics feels a bit "wordy" at first—it’s actually very logical once you see the real-world connections.
1. The Big Picture: Population vs. Sample
Before we can start collecting data, we need to define who or what we are actually interested in. In AQA Mathematics 7357, we use two very specific terms:
The Population
The population represents every single member of the group you want to study. If you want to know the average height of students in your school, the population is every student enrolled at that school.
The Sample
A sample is a smaller group selected from the population. If you only measure the height of 20 students from your school, those 20 students are your sample.
Analogy: Imagine you are cooking a massive pot of vegetable soup. The population is the entire pot of soup. To check if it needs more salt, you don’t eat the whole pot! Instead, you take a single spoonful. That spoonful is your sample.
Why do we use samples?
We use samples because studying a whole population is usually:
1. Too expensive (imagine paying to interview everyone in the UK!)
2. Too time-consuming
3. Impossible (if you are testing the "life span" of lightbulbs, you can't test every bulb because you'd break them all!)
Quick Review Box:
• Population: The whole group.
• Sample: A part of the population.
• Census: When you actually observe every member of a population (like the UK government does every 10 years).
2. Making Inferences
The whole point of taking a sample is to make an inference. This is just a fancy way of saying "making an educated guess about the population based on the sample."
If your spoonful of soup (sample) tastes too salty, you infer that the whole pot (population) is too salty. In your exam, you might be asked to explain what a sample tells us. You should use phrases like "the sample suggests that..." or "we can infer from the sample that..."
Important Point: Different samples can lead to different conclusions! If your sample happens to include only the tallest students in the school, your "inference" about the average height of the whole school will be wrong. This is called sampling error.
3. Sampling Techniques
The AQA syllabus requires you to understand two main ways of picking your sample. How you choose your sample is vital because it determines if your results are biased (unfair or one-sided).
Method A: Simple Random Sampling
In a simple random sample, every member of the population has an equal chance of being selected. It’s like putting everyone’s name in a giant hat and pulling them out one by one.
How to do it (Step-by-Step):
1. Assign a unique number to every member of the population (this list is called a sampling frame).
2. Use a random number generator (on your calculator or computer) to pick the numbers.
3. The people/items corresponding to those numbers are your sample.
Pros: It is completely unbiased. No one is picked because of the researcher's preference.
Cons: You need a full list of the population, which might be hard to get. It can also be impractical if the population is spread over a huge area.
Method B: Opportunity Sampling
Opportunity sampling (sometimes called convenience sampling) is simply picking whoever is available at the time and fits your criteria. For example, if you stand outside a supermarket and ask the first 10 people who walk past to answer a survey, you are using opportunity sampling.
Pros: It is quick, easy, and cheap. You don't need a list of the whole population.
Cons: It is very likely to be biased. If you stand outside a gym, your sample will likely be more "fit" than the general population. It doesn't represent the whole population well.
Memory Aid: Think of Opportunity as "taking the opportunity of whoever is nearby."
4. Critiquing a Sample
In Paper 3, you are often asked to critique (find the flaws in) a sampling method. When you see a question like this, look for two things:
1. Is the sample size large enough?
Small samples are "unreliable." If you ask only 2 people their opinion, one person having a weird opinion changes your results by 50%! A larger sample is usually more representative.
2. Is there Bias?
Think about where and when the data was collected.
Example: If you want to know how much people like football, don't ask people outside a stadium on a Saturday (that’s biased!).
Did you know?
In 1936, a US magazine predicted a landslide victory for Alf Landon in the presidential election based on a sample of 2.4 million people. However, they picked their sample from telephone directories and car registrations. In 1936, only wealthy people had phones and cars. The sample was biased, and Franklin D. Roosevelt actually won in a landslide!
5. Common Mistakes to Avoid
• Confusing Sample and Population: Always read the question carefully. Is the number given the whole group or just the bit they tested?
• Thinking "Random" means "Haphazard": In maths, "random" has a strict definition (equal chance). Picking people on a street corner is not random; it is opportunity sampling.
• Ignoring the Context: If the question is about a "Large Data Set," remember that different samples of that data might show different trends.
Key Takeaways for Section 3.12 K
• The Population is the whole; the Sample is the part.
• Simple Random Sampling gives everyone an equal chance (unbiased but needs a list).
• Opportunity Sampling uses whoever is there (fast but biased).
• Different samples will lead to different inferences.
• To critique a sample, look for bias and size.
Keep practicing! Statistics is all about understanding how we can trust the numbers we see in the news every day. You've got this!