In the previous chapter we presented statistics that can be used to summarize and describe data. Descriptive procedures are not sufficient, however, for testing theories about the effects of treatments or for generalizing relationships from samples to populations. For these purposes, researchers use a process of statistical *inference*. The process of drawing inferences is familiar to everybody. When we decide to read a book by a certain author after having enjoyed other books by that same author, we are inferring something about the probable quality of the new book. When a specific treatment approach produces beneficial effects for a particular patient, a clinician might decide to use that approach for other patients with similar conditions. The difference between these subjective inferences and statistical inference is that the researcher uses objective criteria to make such decisions.

**Inferential statistics** involve a decision making process that allows us to estimate population characteristics from sample data. The success of this process requires that we make certain assumptions about how well the sample represents the larger population. These assumptions are based on two important concepts of statistical reasoning: **probability** and **sampling error**. The purpose of this chapter is to introduce these fundamental concepts and to demonstrate the principles of their application for drawing valid conclusions from research data.

**Probability** is a complex but essential concept for understanding inferential statistics. We all have some notion of what probability means, as evidenced by the use of terms such as "likely," "probably" or "a good chance." We use probability as a means of prediction: "There is a 50% chance of rain tomorrow," or "This operation has a 75% chance of success." Statistically, we can view probability as a system of rules for analyzing a complete set of possible outcomes, or a *sample space*. For instance, a sample space could represent the two sides of a coin or the six faces on a die. An event is a single observable happening or outcome, such as the appearance of tails on the flip of a coin or a 3 on the toss of a die. A sample space could be a set of IQ scores for all students in a given school system. An event might be the random selection of one student's IQ score of 110. In other words, each score in the sample space is a potential event.

*Probability is the likelihood that any one event will occur, given all the possible outcomes.* We use a lowercase *p* to signify probability, expressed as a ratio or decimal. For example, given the two possible outcomes for the flip of a coin, the likelihood of getting tails on any single flip will be 1 of 2, or 1/2, or .5. Therefore, we say that the probability of getting tails is 50%, or *p =* .50. Suppose we want to know the probability of getting a ...