Sampling (Leaving Cert Mathematics): Revision Notes
Sampling
What is a population and sample?
When conducting statistical investigations, researchers often need to study large groups of people or objects. However, studying everyone is usually impractical due to time and cost constraints.
Understanding the distinction between population and sample is fundamental to statistical investigations. This difference determines how we collect data and what conclusions we can draw.
Population: Everything or everybody that could possibly be involved in a particular investigation or study. For example, if you want to investigate whether 14-year-old boys are taller than 14-year-old girls in Ireland, the population would be all 14-year-olds in Ireland (approximately 60,000 people).
Sample: A group selected from the population to collect data. The purpose of a sample is to collect data from some of the population and use it to draw conclusions about the whole population.
Sample size considerations
The size of a sample is crucial for reliable results:
- Too small: The results may not be very reliable or representative of the population
- Too large: The data may take a long time to collect and analyse, making the study impractical
- Just right: Provides reliable results while being manageable to collect and analyse
Finding the optimal sample size is a critical balance. The sample must be large enough to be representative but small enough to be practical. This balance is essential for valid statistical conclusions.
Bias in sampling
Bias occurs when a sample is not properly selected, causing the results to be distorted and not representative of the population as a whole. If bias exists, the conclusions drawn from the sample will not accurately reflect the entire population.
Avoiding Bias is Critical
Bias is one of the most serious threats to valid statistical research. Even with a large sample size, biassed sampling can make results completely unreliable and misleading.
Sources of bias
Bias in a sample may arise from several sources:
1. Choosing a sample which is not representative
This happens when the sampling location or method favours certain types of people. For example, if Cara wants to survey people's attitudes towards gambling and stands outside a casino to question people entering or leaving, her results will be biassed because these people are already involved in gambling.
2. Not identifying the correct population
This occurs when the sample doesn't include all relevant groups. For example, if a school principal wants to find out about students' attitudes to school uniforms but only questions Leaving Certificate students, the results may be biassed because younger students' opinions (from 1st year to 5th year) are not included.
3. Failure to respond to a survey
Many people don't fill in responses to questionnaires sent through the post. Those who do respond may not be representative of the population being surveyed, leading to biassed results.
4. Dishonest answers to questions
People may not provide truthful responses, which can skew the results of the survey.
Example of biassed sampling
Conor wants to find out if people in his town enjoy watching sport. He stands outside a football ground and surveys people's opinions as they go in to watch a match.
Why This Sample is Problematic
This is not a good sample because:
- People who go to watch football matches usually enjoy sport, so the sample may be biassed towards those who like sport
- Generally more men than women go to watch football, so the survey could be gender-biased
Simple random sampling
Simple random sample (commonly called a random sample): A sampling method where every member of the population being considered has an equal chance of being selected. This is one of the best ways to avoid bias in a survey.
Methods for selecting random samples
To choose a simple random sample, you first give each member of the population a number, then select the numbers for the sample using one of these methods:
Random Selection Methods
- Hat method: Put all the numbers into a hat and select however many you need for the sample
- Random number table: Use a table of random numbers to select your sample
- Random number generator: Use a calculator or computer to generate random numbers
Each method ensures equal probability of selection for every population member.
Using random number tables
Random number tables contain sequences of randomly generated numbers that can be used to select unbiased samples.
Worked Example: Football Club Ticket Selection
A football club with 80 members has 5 tickets for an international match. Here are two methods to choose 5 members at random:
Method 1: Hat method
- Give each member a number and write each number on a piece of paper
- Put the pieces of paper into a box and mix them up well
- Choose five pieces of paper
- The members with the five numbers chosen receive the tickets
Method 2: Random number table Using the random number table above:
- Give each of the 80 members a two-digit number, starting at 11 and ending at 90
- Select five two-digit numbers from the random numbers above
- These numbers must be between 11 and 90 inclusive
- Starting at the beginning of the first row and selecting two-digit numbers: 52, 63, 38, 12, 76
- Ignore any number over 90 or any repeated numbers
- The members with these five selected numbers receive the tickets
Using calculators for random numbers
Electronic calculators are very useful for generating random numbers:
Calculator Random Number Steps
- To generate 3-digit numbers, press SHIFT, then press Ran#
- Press = and disregard the decimal point
- If the number displayed is 0.107, write 107
- Press = repeatedly to get more random numbers
This method provides quick and unbiased number generation for sample selection.
Worked examples
Worked Example 1: Identifying Sampling Methods
Question: Amanda wants to choose a sample of 500 adults from the town where she lives. She considers these methods:
- Method 1: Choose people shopping in the town centre on Saturday mornings
- Method 2: Choose names at random from the electoral register
- Method 3: Choose people living in the streets near her house
Answer: Method 2 is most likely to produce an unbiased sample because it gives every adult in the town an equal chance of being selected, making it a simple random sample.
Worked Example 2: Using Random Number Tables
Question: Use the random number table to select a simple random sample of 5 from a population of 50.
Answer:
- Number the population from 01 to 50
- Use two-digit numbers from the table, ignoring numbers over 50 and repeated numbers
- Starting from the beginning: 88 (ignore - over 50), 71 (ignore - over 50), 55 (ignore - over 50), 94 (ignore - over 50), 76 (ignore - over 50)
- Continue: 21, 85 (ignore - over 50), 93 (ignore - over 50), 64 (ignore - over 50), 20, 64 (ignore - repeated)
- Continue until you have 5 valid numbers: 41, 57 (ignore - over 50), 16, 99 (ignore - over 50), 43, 86 (ignore - over 50)
- Sample includes individuals numbered: 21, 20, 41, 16, 43
Worked Example 3: Explaining Bias
Question: Kate stands outside a cinema to survey how often people go to the cinema and how they travel there. Explain why this sample could be biassed.
Answer: This sample is biassed because Kate is only questioning people who are already going to the cinema. These people are likely to go to the cinema more frequently than the general population, so the results won't be representative of everyone's cinema-going habits.
Key Points to Remember:
- Population is the entire group you want to study; sample is the smaller group you actually collect data from
- Sample size matters: too small gives unreliable results, too large is impractical
- Bias makes your results unrepresentative - avoid it by using proper sampling methods
- Simple random sampling gives every member of the population an equal chance of selection
- Random number tables and calculators can help you select unbiased samples