DATA1001 Chap.6 Sampling Distributions
Sampling Distributions
Sampling distributions connect one observed sample to uncertainty about a population. A parameter is a fixed unknown population quantity. A statistic is computed from a sample and varies when the sampling process is repeated. The sampling distribution records that repeated variation. Canvas Topic 7 supplies the chance variability engine, and Topic 8, Sample Surveys, applies it to inference from samples.
For many independent observations, the central limit theorem makes the distribution of a sum or mean approximately Normal around its expected value when the sample is large enough for the population shape. Standard error measures the spread of the statistic across repeated samples.
For a mean it is population SD divided by square root n under the simple independent model, and for a sample proportion the zero and one box gives the familiar square root expression. A confidence interval surrounds an estimate with a margin of error based on standard error and a reference cutoff.
Its confidence level describes long run coverage of the interval procedure, not a probability assigned to the fixed parameter after one interval is calculated. Precision and validity are separate. A large convenience sample may have small model based SE yet remain biased because its sampling frame misses parts of the target population. Increasing n addresses chance variation but not selection, nonresponse or measurement bias.
The bootstrap approximates sampling variation by repeatedly drawing with replacement from the observed sample and recomputing the statistic. It is useful when an analytic SE is awkward, but it does not repair unrepresentative data. A complete exam answer names the population, parameter, sample, statistic, sampling assumptions and interval meaning in context.
What this chapter covers
- 01
Topic 8: Sample Surveys
- 02
Topic 7: Chance Variability for sampling
- 03
Separate parameters from statistics
- 04
Use the central limit theorem with conditions
- 05
Interpret standard error and confidence
- 06
Distinguish precision from sampling bias
Worked example · free
Interpret an interval without moving the parameter
- 1The statistic 0.62 estimates a fixed population proportion.
- 1The interval gives values compatible with the data and method from 0.55 to 0.69.
- 1Across repeated random samples, about 95 percent of intervals built this way would cover the parameter.
- 1The interpretation depends on the sampling frame and response process being credible.
Key terms
- Parameter
- A fixed population quantity that is usually unknown.
- Statistic
- A numerical summary calculated from a sample and used to estimate a parameter.
- Sampling distribution
- The distribution of a statistic across repeated samples generated by the same process.
- Central limit theorem
- A result that makes many sums and means approximately Normal after suitable scaling when sample size is large.
- Confidence interval
- A data based range produced by a method with a stated long run coverage rate.
- Bootstrap
- A resampling method that repeatedly samples with replacement from observed data to approximate statistic variability.
Sampling Distributions FAQ
How do parameters and statistics differ?
A parameter describes the population and is fixed although unknown. A statistic is computed from the sample and changes from sample to sample. Use notation and context carefully so a sample mean is not described as the population mean itself.
What does standard error measure?
Standard error is the standard deviation of a statistic across repeated samples from the same model. It measures sampling variability, not the spread of individual observations. For many means and proportions it shrinks like one over square root n.
What does 95 percent confidence mean?
It describes a procedure that would cover the fixed parameter in about 95 percent of repeated applications under its assumptions. After one interval is observed, avoid saying there is a 95 percent chance that the fixed parameter lies inside it.
Can a narrow interval be biased?
Yes. An ordinary interval can be narrow because model based chance variation is small while the sampling frame or response process systematically misses the target population. Precision does not guarantee validity, so evaluate sampling and measurement before trusting the interval.
What does a bootstrap do?
It repeatedly samples with replacement from the observed data and recomputes the statistic, creating an empirical approximation to sampling variation. It helps when a formula is difficult, but it inherits important limitations of the original sample and does not remove selection bias.
Exam move
Draw three levels for every problem: population, one observed sample and repeated samples. Put the parameter at the population level, the statistic in the observed sample and standard error in the repeated sample distribution. Practise how SE changes when n is multiplied by four, nine or sixteen.
For confidence intervals, perform the calculation and then write the repeated procedure interpretation without assigning probability to a fixed parameter. Pair every precision question with a validity question about the sampling frame, nonresponse and measurement. For bootstrap code, identify what is sampled, why replacement is used and which statistic is recomputed each time.
Work the exam lens under a short timer and check whether each conclusion names the actual target population. If a result comes from volunteers or a restricted frame, state that limitation even when the interval is very narrow. This chapter rewards clear distinctions more than long calculations. Rehearse each conclusion aloud and remove any unsupported claim of certainty.