Iain Pardoe - Applied Regression Modeling

From a statistical perspective, a distribution (strictly speaking, a probability distribution ) is a theoretical model that describes how a random variable varies. For our purposes, a random variable represents the data values of interest in the population, for example, the sale prices of all single‐family homes in our housing market. One way to represent the population distribution of data values is in a histogram, as described in Section 1.1. The difference now is that the histogram displays the whole population rather than just the sample. Since the population is so much larger than the sample, the bins of the histogram (the consecutive ranges of the data that comprise the horizontal intervals for the bars) can be much smaller than in Figure 1.1. For example, Figure 1.2shows a histogram for a simulated population of sale prices. The scale of the vertical axis now represents proportions (density) rather than the counts (frequency) of Figure 1.1.

Figure 1.2Histogram for a simulated population of sale prices, together with a normal density curve.

As the population size gets larger, we can imagine the histogram bars getting thinner and more numerous, until the histogram resembles a smooth curve rather than a series of steps. This smooth curve is called a density curve and can be thought of as the theoretical version of the population histogram. Density curves also provide a way to visualize probability distributions such as the normal distribution. A normal density curve is superimposed on Figure 1.2. The simulated population histogram follows the curve quite closely, which suggests that this simulated population distribution is quite close to normal.

To see how a theoretical distribution can prove useful for making statistical inferences about populations such as that in our home prices example, we need to look more closely at the normal distribution. To begin, we consider a particular version of the normal distribution, the standard normal , as represented by the density curve in Figure 1.3. Random variables that follow a standard normal distribution have a mean of 0 (represented in Figure 1.3by the curve being symmetric about 0, which is under the highest point of the curve) and a standard deviation of 1 (represented in Figure 1.3by the curve having a point of inflection—where the curve bends first one way and then the other—at and ). The normal density curve is sometimes called the “bell curve” since its shape resembles that of a bell. It is a slightly odd bell, however, since its sides never quite reach the ground (although the ends of the curve in Figure 1.3are quite close to zero on the vertical axis, they would never actually quite reach there, even if the graph were extended a very long way on either side).

Figure 1.3Standard normal density curve together with a shaded area of between and , which represents the probability that a standard normal random variable lies between and .

The key feature of the normal density curve that allows us to make statistical inferences is that areas under the curve represent probabilities. The entire area under the curve is one, while the area under the curve between one point on the horizontal axis ( , say) and another point ( , say) represents the probability that a random variable that follows a standard normal distribution is between and . So, for example, Figure 1.3shows there is a probability of 0.475 that a standard normal random variable lies between and , since the area under the curve between and is 0.475.

We can obtain values for these areas or probabilities from a variety of sources: tables of numbers, calculators, spreadsheet or statistical software, Internet websites, and so on. In this book, we print only a few select values since most of the later calculations use a generalization of the normal distribution called the “t‐distribution.” Also, rather than areas such as that shaded in Figure 1.3, it will become more useful to consider “tail areas” (e.g., to the right of point ), and so for consistency with later tables of numbers, the following table allows calculation of such tail areas: Normal distribution probabilities (tail areas) and percentiles (horizontal axis values)

In particular, the upper‐tail area to the right of 1.960 is 0.025; this is equivalent to saying that the area between 0 and 1.960 is 0.475 (since the entire area under the curve is 1 and the area to the right of 0 is 0.5). Similarly, the two‐tail area, which is the sum of the areas to the right of 1.960 and to the left of −1.960, is two times 0.025, or 0.05.

How does all this help us to make statistical inferences about populations such as that in our home prices example? The essential idea is that we fit a normal distribution model to our sample data and then use this model to make inferences about the corresponding population. For example, we can use probability calculations for a normal distribution (as shown in Figure 1.3) to make probability statements about a population modeled using that normal distribution—we will show exactly how to do this in Section 1.3. Before we do that, however, we pause to consider an aspect of this inferential sequence that can make or break the process. Does the model provide a close enough approximation to the pattern of sample values that we can be confident the model adequately represents the population values? The better the approximation, the more reliable our inferential statements will be.