|
Sample size, |
Valid | 30 |
| missing | 0 | |
| Mean | 278.6033 | |
| Median | 278.9500 | |
| Standard deviation | 53.8656 | |
| Minimum | 155.5000 | |
| Maximum | 359.9000 | |
| Percentiles | 25 | 241.3750 |
| 50 | 278.9500 | |
| 75 | 325.8750 |
There are many other methods—numerical and graphical—for summarizing data. For example, another popular graph besides the histogram is the boxplot; see Chapter 6 (www.wiley.com/go/pardoe/AppliedRegressionModeling3e) for some examples of boxplots used in case studies.
1.2 Population Distributions
While the methods of the preceding section are useful for describing and displaying sample data, the real power of statistics is revealed when we use samples to give us information about populations. In this context, a population is the entire collection of objects of interest, for example, the sale prices for all single‐family homes in the housing market represented by our dataset. We would like to know more about this population to help us make a decision about which home to buy, but the only data we have is a random sample of 30 sale prices.
Nevertheless, we can employ “statistical thinking” to draw inferences about the population of interest by analyzing the sample data. In particular, we use the notion of a model—a mathematical abstraction of the real world—which we fit to the sample data. If this model provides a reasonable fit to the data, that is, if it can approximate the manner in which the data vary, then we assume it can also approximate the behavior of the population. The model then provides the basis for making decisions about the population, by, for example, identifying patterns, explaining variation, and predicting future values. Of course, this process can work only if the sample data can be considered representative of the population. One way to address this is to randomly select the sample from the population. There are other more complex sampling methods that are used to select representative samples, and there are also ways to make adjustments to models to account for known nonrandom sampling. However, we do not consider these here—any good sampling textbook should cover these issues.
Sometimes, even when we know that a sample has not been selected randomly, we can still model it. Then, we may not be able to formally infer about a population from the sample, but we can still model the underlying structure of the sample. One example would be a convenience sample—a sample selected more for reasons of convenience than for its statistical properties. When modeling such samples, any results should be reported with a caution about restricting any conclusions to objects similar to those in the sample. Another kind of example is when the sample comprises the whole population. For example, we could model data for all 50 states of the United States of America to better understand any patterns or systematic associations among the states.
Since the real world can be extremely complicated (in the way that data values vary or interact together), models are useful because they simplify problems so that we can better understand them (and then make more effective decisions). On the one hand, we therefore need models to be simple enough that we can easily use them to make decisions, but on the other hand, we need models that are flexible enough to provide good approximations to complex situations. Fortunately, many statistical models have been developed over the years that provide an effective balance between these two criteria. One such model, which provides a good starting point for the more complicated models we consider later, is the normal distribution.
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 1.2 shows a histogram for a simulated population of
Figure 1.2 Histogram for a simulated population of
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 1.3 by the curve being symmetric about 0, which is under the highest point of the curve) and a standard deviation of 1 (represented in Figure 1.3 by the curve having a point of inflection—where the curve bends first one way and then the other—at
