DAO1704 Chap.6 Sampling Distributions and Estimator Precision
Sampling Distributions and Estimator Precision
Statistics vary because samples vary
A probability sample connects a subset to a population through a known selection mechanism. Draw another valid sample and the sample mean usually changes.
The sampling distribution collects the statistic's possible values and probabilities across repeated samples, making uncertainty part of the analysis rather than an afterthought.
Centre and spread are separate estimator properties
An unbiased estimator is centred on the population parameter across repetitions. It can still vary widely in any one sample. Precision concerns that repeat-to-repeat spread.
Among unbiased estimators, smaller variance is preferred, and under the independence model the variance of the sample mean is population variance divided by sample size.
Sample size cannot repair selection
Increasing sample size reduces standard error according to a square-root relationship, but it does not eliminate bias from a voluntary or incomplete sampling mechanism.
The chapter first checks how observations entered the sample, then calculates precision, and finally interprets the estimate for its stated population without calling a single observed statistic exact.
Enumeration keeps the design visible
For a small finite population, list every possible sample and its statistic before applying a shortcut.
The probabilities reveal whether samples are equally likely and whether sampling without replacement creates dependence. Once the exact distribution is visible, expected value and variance become weighted summaries rather than mysterious formulas.
This approach also sharpens interpretation: the centre describes the estimator across repetitions, while the spread records the uncertainty that remains for any one selected sample.
What this chapter covers
- 01
Simple random sampling without replacement
- 02
Statistics as sample-dependent random variables
- 03
Sampling distributions across repeated samples
- 04
Expected value and unbiasedness
- 05
Variance and estimator efficiency
- 06
Standard error and square-root returns
Worked example · free
Enumerating every sample mean
- 2List all six equally likely samples.
- 2Calculate each sample mean.
- 2Obtain the expected sample mean.
- 2Calculate and interpret the variance.
Key terms
- Simple Random Sample
- A probability sample in which every valid sample of the chosen size has equal selection chance.
- Statistic
- A numerical quantity computed from sample observations.
- Sampling Distribution
- The probability distribution of a statistic under repeated samples from the design.
- Unbiasedness
- The property that an estimator's expected value equals its target parameter.
- Estimator Variance
- The repeat-to-repeat variability of an estimator under the sampling design.
- Standard Error
- The standard deviation of the estimator's sampling distribution.
Sampling Distributions and Estimator Precision FAQ
Why is a statistic treated as random?
Its value depends on which units enter the sample. Before selection, several valid samples and corresponding statistics are possible. The sampling distribution represents that design-based uncertainty even though the population parameter itself is fixed.
Does unbiased mean the observed estimate is correct?
No. Unbiasedness concerns the average estimate across repeated samples. A particular sample mean can lie above or below the target. Estimator variance or standard error describes how widely those possible values are dispersed.
Why does a larger voluntary sample remain risky?
Additional voluntary responses can reduce variability within the responding group while preserving selection bias relative to the target population. Check coverage and response mechanisms before applying a precision formula or treating the large count as representative.
Exam move
Start with a finite population small enough to enumerate. List all samples, calculate each statistic and confirm that the probabilities sum to one. Then compute expected value and variance with SUMPRODUCT in Excel and by hand. Change sample size and observe how the distribution contracts.
Finish by contrasting the probability design with a much larger voluntary sample so that selection validity and precision remain separate in your explanation. Create a table of sample sizes and standard errors, then predict the sample-size multiplier needed to halve the error before calculating it. For every result, write one sentence about centring and a separate sentence about spread.
Keeping those sentences apart prevents unbiasedness from being mistaken for certainty. Close the session by identifying the population, sampling frame and selection mechanism in a real survey. Do not calculate precision until all three can be named without guesswork.
State which missing group would most threaten the conclusion and why its omission would not be repaired by collecting more responses from the groups already represented.
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