Welcome to Sampling
Welcome to the chapter on Sampling for CCEA Further Mathematics (AS 2 Section C: Statistics). Whether you love statistics or find it a little intimidating, don't worry! This topic is very practical and intuitive. In the real world, we rarely have the time or money to measure every single object or ask every single person a question. Sampling is the art and science of gathering just enough data to make accurate, reliable predictions about a whole group.
1. Populations, Censuses, and Samples
Key Definitions
Let's start by looking at the basic vocabulary of sampling:
• Population: The entire collection of all individuals, items, or units that you are interested in investigating. For example, all students enrolled in Northern Ireland schools, or every lightbulb produced in a factory today.
• Census: An investigation where information is gathered from every single member of the population.
• Sample: A selection of individual units chosen from the population to represent the whole group.
• Sampling Unit: An individual member or element of the population that can be sampled (e.g., one student, one lightbulb).
• Sampling Frame: A complete, numbered list or register of all the sampling units in the population from which the sample is drawn (e.g., a school register or an electoral roll).
Analogy time: Imagine you are cooking a giant pot of soup. You don't need to drink the entire pot to find out if it needs more salt. You just stir it thoroughly and taste a single spoonful! Here, the whole pot is the population, drinking the entire pot is a census, and your spoonful is the sample.
Census vs. Sample: Pros and Cons
Why not always do a census? Let's compare:
Census:
• Advantages: It provides completely accurate and unbiased results since every single unit is measured. There is no sampling variation.
• Disadvantages: It can be extremely time-consuming and expensive. It is impractical for large populations. It cannot be used if the testing process is destructive (for instance, if you are testing the lifespan of batteries or crash-testing cars, a census would destroy all your products!).
Sample:
• Advantages: Much quicker, cheaper, and easier to administer than a census. It is the only practical option when testing destroys the items.
• Disadvantages: Data from a sample may not be completely representative, introducing sampling error. Potential bias can arise if the sample is poorly chosen.
Key Takeaway: Use a census when the population is small or when 100% accuracy is essential. Use a sample when the population is large, resources are limited, or testing is destructive.
2. Random (Probability) Sampling Methods
In a random sample, every item in the population has a known, non-zero probability of being selected. This eliminates subjective bias from the researcher.
Method A: Simple Random Sampling (SRS)
In a simple random sample of size \(n\), every individual unit has an equal chance of being selected, and every possible combination of \(n\) units has an equal chance of being the sample chosen.
How to carry it out:
1. Allocate a unique identifying number from \(1\) to \(N\) to each member of the sampling frame.
2. Generate \(n\) distinct random numbers between \(1\) and \(N\) using a random number generator, calculator, or random number table.
3. Select the corresponding units from the sampling frame, ignoring any duplicate numbers generated.
• Pros: Completely unbiased; every member has an equal chance.
• Cons: Requires a complete sampling frame; can be difficult and expensive to carry out if the population is spread across a large geographical area.
Method B: Systematic Sampling
In systematic sampling, elements are chosen at regular intervals from an ordered list.
How to carry it out:
1. Number the items in the sampling frame from \(1\) to \(N\).
2. Calculate the sampling interval \(k = \frac{N}{n}\) (rounded to the nearest whole number).
3. Choose a random starting point between \(1\) and \(k\). Let this be \(r\).
4. Select the units numbered \(r, r + k, r + 2k, r + 3k, \dots\) until your sample of size \(n\) is complete.
• Pros: Simple and quick to use; suitable for large populations.
• Cons: Can introduce severe bias if the list has a hidden cyclic or periodic pattern that matches the interval \(k\).
Method C: Stratified Sampling
In stratified sampling, the population is divided into mutually exclusive sub-groups called strata (e.g., based on age, year group, or gender) that are important to the investigation. A simple random sample is then taken from each stratum in proportion to the stratum's size in the overall population.
Formula for Stratified Sample Size:
The number of items \(n_i\) to select from stratum \(i\) of size \(N_i\) is given by:
\(n_i = \frac{N_i}{N} \times n\)
where \(N\) is the total population size and \(n\) is the total required sample size.
Worked Example:
A school has \(600\) students: \(300\) in Key Stage 3, \(200\) in Key Stage 4, and \(100\) in Sixth Form. A stratified sample of size \(n = 60\) is required.
• Key Stage 3: \(\frac{300}{600} \times 60 = 30\) students
• Key Stage 4: \(\frac{200}{600} \times 60 = 20\) students
• Sixth Form: \(\frac{100}{600} \times 60 = 10\) students
Then, within each group, students are selected using simple random sampling.
• Pros: Guarantees representation of key sub-groups; yields more precise estimates if strata are homogeneous.
• Cons: The population must be easily classified into distinct strata, and a full sampling frame is still required for each stratum.
Key Takeaway: Random sampling methods (Simple Random, Systematic, Stratified) eliminate researcher bias, but they all require an accurate sampling frame.
3. Non-Random (Non-Probability) Sampling Methods
In non-random sampling, units do not have a known chance of being selected. These methods do not strictly require a formal sampling frame.
Method A: Quota Sampling
An interviewer selects an assigned number (quota) of people from different demographic groups (e.g., \(20\) males over 50, \(20\) females under 30) until each quota is filled. The interviewer chooses who to interview.
• Pros: No sampling frame needed; quick, easy, and inexpensive; ensures specific groups are included.
• Cons: Introduces interviewer bias (e.g., choosing friendly-looking people); non-respondents are simply replaced rather than recorded, which can mask bias.
Method B: Opportunity (Convenience) Sampling
A sample is taken from people or items that are readily available at the time of the study (e.g., surveying the first \(30\) people who walk past you outside a shopping centre on a Tuesday morning).
• Pros: Very easy, fast, and cheap to execute.
• Cons: Highly unrepresentative and prone to substantial bias (e.g., surveying outside a shop on a Tuesday morning excludes people working standard office hours).
Key Takeaway: Non-probability methods (Quota and Opportunity) are cheap and do not require a list of the population, but they carry a high risk of bias.
4. Parameters, Statistics, and Estimators
In Further Mathematics, we often explore the theoretical links between samples and the populations they represent.
Population Parameter vs. Sample Statistic
• Parameter: A fixed numerical value that describes a characteristic of the whole population (e.g., population mean \(\mu\), population variance \(\sigma^2\), or population proportion \(p\)). Parameters are usually unknown constants.
• Statistic: A numerical quantity calculated purely from the observations in a sample, containing no unknown parameters (e.g., sample mean \(\bar{X}\), sample variance \(S^2\)). A statistic varies from sample to sample, meaning it is a random variable!
Memory Trick: Parameter describes the Population; Statistic describes the Sample.
Sampling Distributions
Because a statistic is a random variable, it has its own probability distribution. This is called the sampling distribution of the statistic. It represents all possible values the statistic could take across all possible samples of size \(n\), along with their associated probabilities.
Unbiased Estimators
An estimator \(T\) is said to be an unbiased estimator of a population parameter \(\theta\) if its expected value is equal to that parameter:
\(\text{E}(T) = \theta\)
For a random sample \(X_1, X_2, \dots, X_n\) drawn from a population with mean \(\mu\) and variance \(\sigma^2\):
1. Sample Mean \(\bar{X}\):
The sample mean is defined as \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\).
Its expected value is:
\(\text{E}(\bar{X}) = \mu\)
Therefore, \(\bar{X}\) is an unbiased estimator of the population mean \(\mu\).
2. Sample Variance and Unbiased Estimator \(S^2\):
If we simply calculated the mean squared deviation from the sample mean using division by \(n\), it would, on average, underestimate the true population variance \(\sigma^2\).
To correct for this bias, we divide by \(n - 1\) instead of \(n\):
\(S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X})^2 = \frac{1}{n-1}\left(\sum X_i^2 - \frac{(\sum X_i)^2}{n}\right)\)
Because \(\text{E}(S^2) = \sigma^2\), \(S^2\) is the unbiased estimator of the population variance.
Standard Error of the Mean
The variance of the sample mean is given by:
\(\text{Var}(\bar{X}) = \frac{\sigma^2}{n}\)
The standard deviation of the sampling distribution of \(\bar{X}\) is called the Standard Error (SE):
\(\text{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}}\)
Notice: As the sample size \(n\) increases, the standard error decreases. This means larger samples provide more precise estimates of the population mean!
Key Takeaway: A sample statistic is a random variable. \(\bar{X}\) is an unbiased estimator of \(\mu\), and \(S^2\) (with divisor \(n-1\)) is an unbiased estimator of \(\sigma^2\).
5. Common Mistakes to Avoid
• Confusing Sampling Frame with Population: A population is the collection of items itself; a sampling frame is the actual literal list/database of names or numbers used to draw the sample.
• Dividing by \(n\) instead of \(n-1\) for Unbiased Variance: Remember that when calculating an unbiased estimate of the population variance from sample data, always divide by \(n-1\).
• Thinking Systematic Sampling is Simple Random Sampling: In systematic sampling, once the first item is chosen, all other selections are fixed. Not all combinations of \(n\) units are possible, so it is not an SRS.
• Forgetting Units or Context: In exam questions asking for advantages or disadvantages, always link your answer back to the specific real-world context given in the question.
Quick Review Summary
• Census: Every member tested (accurate, but expensive and impossible if destructive).
• Sample: Subset tested (quicker and cheaper, but subject to sampling error).
• Simple Random: Equal chance for every combination; requires a sampling frame.
• Systematic: Every \(k\)-th item chosen from a list.
• Stratified: Proportional selection from distinct strata: \(n_i = \frac{N_i}{N} \times n\).
• Quota / Opportunity: Non-random methods; convenient, but prone to bias.
• Unbiased Estimators: \(\text{E}(\bar{X}) = \mu\) and \(\text{E}(S^2) = \sigma^2\) where divisor is \(n-1\).
• Standard Error: \(\text{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}}\).