Describing the Population

Summary

An “object” is in this book something which has to be manufactured (for products) or more generally realized by using a specific process. So the word “object” must be understood in a very large sense: it can be a part of an equipment (such a mechanical part, or an electronic board, or even an electronic chip), the equipment itself (such as a printer, or a reactor), the software, the system it belongs to (such as an airplane), a building or part of it, a trench, a tunnel, etc. or even an activity (such as boring a hole in a plate, painting a room or performing a surgical operation).

The term “population” is generally used for naming the whole set of objects the statistician works on. The same name is kept in this book; however, our populations are very special: a population is the set of objects which constitute what will be called a “product family”. Such a family must be as homogeneous as possible, the degree of homogeneity being let to the cost analyst.

In a product family, objects may more or less differ, depending on the level of homogeneity. Their differences are quantified, or more generally described, by variables. A rule of the art is “the less homogeneous the product family, the more variables you need”. At the minimum, the size of the objects — by their physical size, or by their functional size — must be described.

This chapter first presents a few definitions.

The purpose of this book is to forecast something about any object of a product family (our population): it can be the cost of manufacturing it, or the time to do it, or the tooling which is needed, or anything else. In order to be able to make this forecast, we get a sample (it is the data we start from) from which we are going to extract the information we need. The logic for doing it is exposed in Section 4 of Chapter 1 which is an important section of this book: it shows how the study of a complex distribution of several variables can be solved by studying the distribution of one variable only.

The concept of distribution of one variable will therefore be present in any part of this book: it must be fully understood by the user and, for this reason, this chapter presents different ways for describing such a distribution: the purpose of this description is that it would be extremely difficult to continuously work with the full distribution: it is much more easier to use a limited set of descriptors.

The next chapter is devoted to the description of several “standard” distributions, which are very well known: if any of our distributions looks similar to one of them, solutions are immediately available.

Distributions of values of one variable only play an important role in this book: as explained in the previous chapter, what we are trying to do is to replace complex distributions involving several variables by a mono-variable distribution.

Several characteristics of these distributions will be frequently used. We found it convenient to group them in one chapter.

A distribution is defined here as a set of I values that takes a variable here called z. These values will be called z₁, z₂, ..., z_i, ..., z_I.

Such a set is difficult to handle. To simplify computations based on the sample, mathematicians found a few concepts that can be substituted to it. The purpose of these concepts is to describe the distribution by a few characteristics; for most of the applications, these few characteristics describe this distribution with a sufficient accuracy for practical purposes. Therefore, most of the computations can be made on them instead of the distribution itself.

There are two ways to build these concepts. The first one can be defined as “analytic”, the second one as “global”. There are obviously relationships between these two ways.