Introduction to Statistical Thinking

In the previous chapters we examined the variability in data. In the statistical context, data is obtained by selecting a sample from the target population and measuring the quantities of interest for the subjects that belong to the sample. Different subjects in the sample may obtain different values for the measurement, leading to variability in the data.

This variability may be summarized with the aid of a frequency table, a table of relative frequency, or via the cumulative relative frequency. A graphical display of the variability in the data may be obtained with the aid of the bar plot, the histogram, or the box plot.

Numerical summaries may be computed in order to characterize the main features of the variability. We used the mean and the median in order to identify the location of the distribution. The sample variance, or better yet the sample standard deviation, as well as the inter-quartile range were all described as tools to quantify the overall spread of the data.

The aim of all these graphical representations and numerical summaries is to investigate the variability of the data.

The subject of this chapter is to introduce two other forms of variability, variability that is not associated, at least not directly, with the data that we observe. The first type of variability is the population variability. The other type of variability is the variability of a random variable.

The notions of variability that will be presented are abstract, they are not given in terms of the data that we observe, and they have a mathematical-theoretical flavor to them. At first, these abstract notions may look to you as a waste of your time and may seem to be unrelated to the subject matter of the course. The opposite is true. The very core of statistical thinking is relating observed data to theoretical and abstract models of a phenomena. Via this comparison, and using the tools of statistical inference that are presented in the second half of the book, statisticians can extrapolate insights or make statements regarding the phenomena on the basis of the observed data. Thereby, the abstract notions of variability that are introduced in this chapter, and are extended in the subsequent chapters up to the end of this part of the book, are the essential foundations for the practice of statistics.

The first notion of variability is the variability that is associated with the population. It is similar in its nature to the variability of the data. The difference between these two types of variability is that the former corresponds to the variability of the quantity of interest across all members of the population and not only for those that were selected to the sample.

In Chapters Chapter 2 and Chapter 3 we examined the data set “ex.1” which contained data on the sex and height of a sample of 100 observations. In this chapter we will consider the sex and height of all the members of the population from which the sample was selected. The size of the relevant population is 100,000, including the 100 subjects that composed the sample. When we examine the values of the height across the entire population we can see that different people may have different heights. This variability of the heights is the population variability.

The other abstract type of variability, the variability of a random variable, is a mathematical concept. The aim of this concept is to model the notion of randomness in measurements or the uncertainty regarding the outcome of a measurement. In particular we will initially consider the variability of a random variable in the context of selecting one subject at random from the population.

Imagine we have a population of size 100,000 and we are about to select at random one subject from this population. We intend to measure the height of the subject that will be selected. Prior to the selection and measurement we are not certain what value of height will be obtained. One may associate the notion of variability with uncertainty — different subjects to be selected may obtain different evaluations of the measurement and we do not know before hand which subject will be selected. The resulting variability is the variability of a random variable.

Random variables can be defined for more abstract settings. Their aim is to provide models for randomness and uncertainty in measurements. Simple examples of such abstract random variables will be provided in this chapter. More examples will be introduced in the subsequent chapters. The more abstract examples of random variables need not be associated with a specific population. Still, the same definitions that are used for the example of a random variable that emerges as a result of sampling a single subject from a population will apply to the more abstract constructions.

All types of variability, the variability of the data we dealt with before as well as the other two types of variability, can be displayed using graphical tools and characterized with numerical summaries. Essentially the same type of plots and numerical summaries, possibly with some modifications, may and will be applied.

A point to remember is that the variability of the data relates to a concrete list of data values that is presented to us. In contrary to the case of the variability of the data, the other types of variability are not associated with quantities we actually get to observe. The data for the sample we get to see but not the data for the rest of the population. Yet, we can still discuss the variability of a population that is out there, even though we do not observe the list of measurements for the entire population. (The example that we give in this chapter of a population was artificially constructed and serves for illustration only. In the actual statistical context one does not obtain measurements from the entire population, only from the subjects that went into the sample.) The discussion of the variability in this context is theoretical in its nature. Still, this theoretical discussion is instrumental for understanding statistics.