QCE Mathematical Methods - Unit 4 - Sampling and proportions
Random Sampling | QCE Mathematical Methods
Learn QCE Mathematical Methods random sampling, bias, procedures for randomness and why samples vary.
Updated 2026-05-18 - 5 min read
QCAA official coverage - Mathematical Methods 2025 v1.3
Exact syllabus points covered
- Understand the concept of a random sample.
- Understand sources of bias in samples, and procedures to ensure randomness.
- Identify and use procedures to ensure randomness.
- Recognise and use graphical displays of real and simulated data of random samples from various types of distributions, including uniform, Bernoulli, binomial and normal.
Statistics is often about learning something about a population without measuring every member of that population. A sample is the subset you actually observe. Random sampling is how you try to make that subset fair enough to support inference.
Population and sample
The population is the whole group you want to make a statement about. The sample is the group you collect data from.
For example, if you want to estimate the proportion of Year 12 students in Queensland who study Mathematical Methods, the population is not just your school. A sample from one Methods class would be convenient, but it would not represent the full target population.
Original Sylligence diagram for sampling variability displays.
A simple random sample is one where every member of the population has a known and equal chance of being selected, and the selection is driven by a random process rather than convenience. In practice, questions may use random-number generators, shuffled labels, computer simulations or tables of random digits.
Random does not mean "spread out nicely". A random sample can still accidentally contain more of one subgroup than another. The point is that the procedure is fair before the data are collected.
Bias
Bias is a systematic distortion in the sample. It does not disappear just because the sample is large.
Common sources include:
- convenience sampling
- voluntary response
- non-response
- wording that pushes respondents toward an answer
- sampling only at one location or one time
- excluding groups that are hard to reach
If a council surveys people at a shopping centre on a weekday morning, the sample may under-represent people at work, students in class and people who shop online.
Here is how common bias sources affect interpretation:
| Bias source | What can go wrong | | --- | --- | | Convenience sampling | easy-to-reach people may not represent the population | | Voluntary response | people with strong opinions are more likely to answer | | Non-response | selected people who do not reply may differ from those who do | | Leading wording | the question nudges respondents toward one answer | | Undercoverage | part of the population has little chance of selection |
Procedures for randomness
Randomness can be improved by:
- using a random number generator
- sampling from a complete list
- stratifying when important subgroups need representation
- avoiding self-selected respondents
- making the data collection process consistent
If important subgroups exist, stratified random sampling can be more appropriate than one simple random sample. For example, if a school wants opinions across year levels, it can randomly sample within each year level so Year 10, Year 11 and Year 12 are all represented.
Sample variability
Even with a good random process, samples vary. Two random samples from the same population will not usually give the same result. That variability is not a failure; it is the reason we need distributions, standard deviations and confidence intervals.
Graphical displays such as dot plots, histograms and simulated sampling distributions help show this variation.
Methods expects you to read both real and simulated displays. For example:
- a dot plot of repeated sample means shows how samples vary around the population mean
- a histogram of simulated binomial counts shows the likely number of successes in repeated sets of trials
- a histogram of simulated sample proportions shows how $\hat{p}$ centres near $p$
- a normal curve overlay may be reasonable for large samples, but only after checking the context
Simulations are especially useful when a formula is hard to feel intuitively. If a population has success probability $p=0.40$, you can simulate many samples of size $n=50$, calculate $\hat{p}$ for each one, then inspect the distribution of those $\hat{p}$ values. The centre should sit near $0.40$, while the spread shows sampling variation.
Worked example
Worked simulation interpretation
Common mistake
Also be careful with the phrase "representative sample". A sample is not representative just because it has many people. The sampling process must give the target population a fair chance to appear in the sample.