ECMT1020 Chap.6 Samples, Statistics and Estimators
Samples, Statistics and Estimators
Sampling connects an unknown population to observed data. This chapter distinguishes parameters, statistics, estimators and estimates, then treats an estimator as a random variable with its own sampling distribution. It calculates sample means and corrected sample variances, explains why the n minus one denominator appears, and separates sampling variation from measurement error or other data-quality problems.
Before the sample is observed, an estimator can take different values and therefore has a probability distribution. After observation, the same rule produces one estimate. Repeated-sample thinking explains why two competent researchers can obtain different estimates without either making a mistake. It also clarifies the role of random sampling assumptions, which describe how observations relate to a common population.
Sample size counts rows, but the effective information can be smaller when observations share entities, dates or common shocks.
What this chapter covers
- 01
Populations and samples
- 02
Parameters and statistics
- 03
Estimators and estimates
- 04
Sampling distributions
- 05
Sample means
- 06
Corrected sample variance
Worked example · free
Calculate two sample summaries
- +1The total is 60 and n is five, so the mean is 12.
- +1Deviations are 0, 3, -3, 2 and -2.
- +1Squared deviations sum to 26.
- +1Divide by n-1=4 to obtain 6.5.
- +1The mean has minute units and the variance has squared-minute units.
Key terms
- Population
- The full collection targeted by an empirical question.
- Sample
- The observed subset used to learn about a population.
- Parameter
- A fixed numerical feature of a population.
- Statistic
- A function of sample observations containing no unknown parameter.
- Sampling distribution
- The distribution of a statistic across possible samples.
- Sampling variation
- Changes in a statistic caused by drawing different samples through the same procedure. It exists even under perfect measurement and is represented by the estimator's sampling distribution rather than by an error in one recorded observation.
- Variance correction
- The use of n minus one in the usual sample-variance estimator after the sample mean has been estimated. The denominator accounts for the zero-sum restriction on deviations and removes a systematic downward bias.
- Random sampling
- A sampling benchmark under which selection follows a stated random mechanism. Its assumptions justify repeated-sample calculations but must be checked against clustering, selection and time ordering.
Samples, Statistics and Estimators FAQ
What is the difference between an estimator and estimate?
An estimator is the rule applied to a random sample. An estimate is the numerical value produced after a particular sample has been observed and substituted into that rule.
Why divide sample variance by n minus one?
Estimating the sample mean imposes a zero-sum restriction on deviations. The correction removes the downward bias that division by n would create for population-variance estimation.
Is sampling variation measurement error?
No. Different random samples vary even with perfect measurement. Measurement error concerns inaccurate recorded variables and is a separate source of uncertainty or bias in the empirical analysis.
Exam move
Write the population parameter, estimator rule and realised estimate on separate lines. Drill sample mean and variance calculations by listing deviations and their units. Then imagine repeating the sampling procedure and describe what changes across samples. That thought experiment makes standard errors, bias and precision easier to understand in the next chapters. Start each exercise with an object audit.
Under “fixed”, write the population and its parameter. Under “random before sampling”, write the observations and estimator. Under “observed now”, write the numerical estimate. Use uppercase or symbolic notation consistently so those roles do not merge.
For sample means, add the values once and divide once; for sample variance, calculate the mean, list deviations, verify their sum is zero, square them and divide by n minus one. Preserve units on each line. The zero-sum deviation check detects a wrong mean before it affects every square. Practise explaining the correction rather than reciting it.
Estimating the mean imposes one linear restriction, leaving n minus one deviations free to vary. Contrast the corrected estimator with division by n and ask which target each quantity serves. Add sampling-distribution sketches to numerical work. Imagine drawing many samples of the same size, mark the estimate from each, and describe the centre and spread.
Then change sample size and predict which feature should narrow under the stated assumptions. Keep sampling variation distinct from measurement error by writing one example of each for the same variable. A new random sample changes because different units are selected; a measurement error changes because recorded values do not equal the intended quantities.
Larger samples address the first under suitable conditions but may preserve systematic recording problems. Review questions by identifying their target: population mean, population variance, proportion or coefficient. A statistic can be computed correctly yet be poorly chosen for that target.
Finish every solution with the parameter being estimated and the estimate's units, then note any dependence or selection feature that makes the simple random-sample benchmark questionable.
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