Population, sample and statistical language
The population is the full target group about which an inference is made. The sample is the group actually measured, so its numerical summary is a statistic used to estimate a population parameter
\[N=\text{population size},\quad n=\text{sample size}\]
- Name the population The population is every unit in the target group of the study \(\displaystyle \text{population}=\text{all }240\text{ households},\quad N=240\)
- Name the sample The sample is the subset of units whose results are recorded \(\displaystyle \text{sample}=30\text{ households},\quad n=30\)
- Use the source group A summary from the full population is a parameter, while a summary from the sample is a statistic \(\displaystyle \frac{21}{30}\text{ from the sample}\Rightarrow\text{sample statistic}\)
- Read the selection procedure The sampling method is identified by how units are selected, not by the study topic \(\displaystyle \text{random start and every }5\text{th unit}\Rightarrow\text{systematic sampling}\)
Worked example
A quality check concerns 480 panels. A simple random sample of 60 panels is measured, and 21 meet the stated standard. Identify the population and sample, then classify the reported proportion
\[N=480,\quad n=60,\quad\widehat p=\frac{21}{60}=\frac{7}{20}\]
- \(\displaystyle N=480\)The population is all 480 panels in the target production group
- \(\displaystyle n=60\)The sample is the 60 panels that were measured
- \(\displaystyle \widehat p=\frac{21}{60}=\frac{7}{20}\)The reported proportion summarises the measured sample
- \(\displaystyle \frac{21}{60}\Rightarrow\text{sample statistic}\)A summary calculated from a sample is a statistic rather than a population parameter
Common mistakes
Swapping population and sample
✕\(\displaystyle \text{population}=\text{the }60\text{ measured panels},\quad\text{sample}=\text{all }480\text{ panels}\)✓\(\displaystyle \text{population}=\text{all }480\text{ panels},\quad\text{sample}=\text{the }60\text{ measured panels}\)
The population is the full target group and the sample is the measured subsetCalling a sample summary a parameter
✕\(\displaystyle \frac{21}{60}\Rightarrow\text{population parameter}\)✓\(\displaystyle \frac{21}{60}\Rightarrow\text{sample statistic}\)
The source group determines the classification, and this value comes from the measured sampleAccepting self-selection as random sampling
✕\(\displaystyle \text{self-selected responses}\Rightarrow\text{simple random and representative}\)✓\(\displaystyle \text{self-selected responses}\Rightarrow\text{not necessarily representative}\)
When units choose whether to respond, the respondents can differ from the target populationDifficulty levels
introIdentify the population, sample and sample size from a simple description
standardClassify a numerical summary as a population parameter or sample statistic
challengeIdentify a sampling method and judge whether the sample is likely representative
examClassify a summary and select the reason based on its source group
Questions about this topic
What is the difference between a population and a sample?+
The population is the full target group, while the sample is the subset that is actually measured
What is the difference between a parameter and a statistic?+
A parameter summarises a population and a statistic summarises a sample
Why is the selection procedure important?+
The procedure determines the sampling method and affects whether the measured sample is likely to reflect the population
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 14: Statistical inference
In this chapter
Populations, samples, parameters, statistics and sampling methodsAllocate a stratified random sampleFind a percentage of a quantityCalculate a sample proportion and list its possible valuesExact sampling distribution of a sample proportionFinite events, probability assignments and complementsConditional probability from exact representationsEvaluate factorial and combination expressionsConvert between fractions, decimals and percentagesLarge-sample conditions and the sample-proportion distributionCalculate a standard score or actual valueUse z-score boundaries to estimate normal-model percentages and countsCalculate the standard deviation from the formulaSolve a linear equationSolve a quadratic exactly with the quadratic formulaApproximate confidence intervals for a population proportionRearrange literal equations and recognise equivalent line formsInterpret simulated sampling distributions and interval variationDesign and interpret a probability simulation