Whole group vs part
| Term | Meaning | Example |
|---|---|---|
| Population | everyone or everything of interest | all high school students in Ontario |
| Sample | the part actually studied | 800 randomly chosen Ontario students |
| Census | data from the whole population | Statistics Canada’s national census |
Parameter vs statistic
| Describes | Symbols | Usually | |
|---|---|---|---|
| Parameter | a population | μ (mean), σ (standard deviation), p (proportion) | unknown |
| Statistic | a sample | x̄ (mean), s (standard deviation), p̂ (proportion) | calculated |
Memory aid: Parameter–Population; Statistic–Sample.
We use statistics to estimate parameters.
Example
A poll asks 1,200 Canadian adults whether they support a policy; 54% say yes.
- Population: all Canadian adults
- Sample: the 1,200 surveyed
- Statistic: p̂ = 0.54
- Parameter: the true proportion p of all adults (unknown)
Why not always a census?
| Census | Sample |
|---|---|
| exact (if done well) | estimate with some uncertainty |
| expensive and slow | cheaper and faster |
| sometimes impossible | practical |
| necessary for destructive testing |
Destructive testing: testing every light bulb to failure would leave none to sell. A sample is the only option.
Sampling variability
Different random samples from the same population give slightly different statistics. This sampling variability is natural and predictable.
Example: Five random samples of 100 voters might give 52%, 55%, 49%, 53%, 54% support. They cluster around the true value.
Larger samples vary less, which is why big random samples give more precise estimates.
Representative samples
A sample is useful only if it represents the population. A huge sample from the wrong group can mislead. (In a famous 1936 US election poll, millions of responses still predicted the wrong winner, because the sample came mostly from wealthier households.)
Worked examples
Example 1: A school surveys 60 of its 900 students about lunch. Population? All 900 students. Sample? The 60 surveyed.
Example 2: "The average height of all Grade 12 students in Canada" is a parameter or statistic? Parameter.
Example 3: "The mean of 50 measured bolts is 3.02 cm" is a? Statistic.
Common mistakes
- Mixing up parameter and statistic.
- Thinking a sample result equals the population value exactly.
- Thinking bigger always means better, regardless of how the sample was chosen.
Try it yourself
- A factory tests 200 of 50,000 phones. Identify the population and sample.
- Is Statistics Canada’s national census a census or a sample?
- Is p̂ a parameter or a statistic?
- Why would five samples of the same size give different means?
Answers: 1) population: all 50,000 phones; sample: the 200 tested 2) a census 3) a statistic 4) sampling variability
Quick recap
- A population is the whole group; a sample is part of it; a census studies everyone.
- Parameters describe populations; statistics describe samples.
- Sampling variability is normal; larger random samples vary less.