8. Estimating unknown quantities from a sample

This chapter delves into the distinction between descriptive and inferential statistics, focusing on the latter’s aim of deriving knowledge about unknown population parameters from observed data. It introduces estimation theory, the first of two primary pillars of inferential statistics, following a foundational discussion on sampling theory. The chapter begins by exploring key concepts such as populations, samples, and the importance of sampling methods, distinguishing between random and biased sampling. Practical sampling methods such as simple random sampling, stratified sampling, snowball sampling, and convenience sampling are examined, with emphasis on their implications for statistical inference. The chapter underscores the criticality of understanding these concepts for designing studies and making valid inferences, noting that in real-world research, truly random samples are rare.

Building on this, the chapter explains the mathematical underpinnings of estimation through the law of large numbers and the central limit theorem. These principles demonstrate how sample statistics approximate population parameters as sample sizes increase, and why the sampling distribution of the mean becomes normal irrespective of the population’s initial distribution. The chapter also discusses practical techniques for estimating population means, variances, and standard deviations, noting common biases and how to adjust for them. Finally, it introduces confidence intervals as a way to express the uncertainty associated with parameter estimates, distinguishing between frequentist and Bayesian interpretations. The content provides a robust framework for understanding how statisticians use data to estimate population characteristics and addresses foundational tools essential for applied research.