Sampling and Reality

Abstract

At the beginning of Chapter 7 I asserted that sampling was at the very heart of the statistical principles applied in this book. I hope the chapters that lie between there and here have made clearer just what that means. Whether the task is estimating the population mean or proportion, comparing means in several batches, comparing proportions in several batches, or investigating the relationship between two measurements, the logic of the approaches statisticians take involves thinking about the batches of numbers we are working with as samples from a larger population. It is this larger population that really interests us. Sometimes this is literally and obviously true. If, for example, we excavate an entire rock shelter site and recover 452,516 pieces of lithic debitage, we might well select some kind of random sample of this debitage for detailed analysis with the objective of characterizing the entire population of 452,516 waste flakes. In this case we would have a sample of waste flakes that we would use to make statements about the population of all debitage at the site from which the sample was selected. The sampling design we used might well be rather complicated. For instance, we might want to be able to compare one stratum in the site to the others, so we might separately select a sample from each stratum. The techniques discussed in Chapters 9–11 would enable us to determine approximately how large each of these samples would need to be in order to accomplish our aims, and they would enable us to estimate means of measurements we might make and the proportions of different categories we might define in the several populations consisting of debitage from each stratum. We could attach error ranges to these estimates that would help us to know at what confidence level and with what precision we could discuss these means and proportions (Chapters 9–11). We could compare the means of the measurements in different strata using these estimates and error ranges or using t tests and analysis of variance (Chapters 15 and 13). We could compare the proportions of the categories in different strata using the estimates and error ranges or using chi-square (Chapter 14). We could evaluate the strength and significance of relationships between measurements with a regression analysis (Chapter 15). If we had ranks rather than true measurements, we could use a rank correlation (Chapter 16). We could combine samples from different strata to say things about the debitage from the site as a whole (Chapter 17). If there were some category of material that just did not appear in our sample, we could evaluate the confidence with which we could talk about its rarity in the population from which the sample came (Chapter 19). All these analyses would provide us with ways to say how much confidence we had that some observation of interest in the sample reflected something that occurred in the population from which the sample was selected as well. This is a very straightforward application of the principles in Chapters 9–18.