16.1 Descriptive and Inferential Statistics

There are two basic types of statistics: descriptive and inferential. With descriptive statistics, researchers summarize the information they have gleaned in their studies. The raw data can be organized and presented through tables, graphs, and charts, some of which we present in this appendix. We can also use descriptive statistics to represent the average and the spread of the data (how dispersed the values are), a topic to which we will return later. The goal of descriptive statistics is to describe the data, or offer a snapshot of what researchers have observed in their study. Inferential statistics, on the other hand, goes beyond simple data presentation. With inferential statistics, for example, we can determine the probability of events and make predictions about general trends. The goal of inferential statistics is to generalize findings from studies, make predictions based on relationships among variables, and test hypotheses. Inferential statistics also can be used to make statements about how confident we are in our findings based on the data collected.

In Chapter 1, we defined a hypothesis as a statement used to test a prediction. Once a researcher develops a hypothesis, she gathers data and uses statistics to test it, an important component of inferential statistics. Hypothesis testing involves mathematical procedures to determine whether the data support the hypothesis or if they simply result from chance. Let’s look at an example to see how this works. (And you might find it useful to review Chapter 1 if your knowledge of research methods is a little rusty.)

Suppose a researcher wants to determine whether taking vitamin D supplements can boost cognitive function. The researcher designs an experiment to test if giving participants vitamin D pills (the independent variable) leads to better performance on some sort of cognitive task, such as a memory test (the test score is the dependent variable). Participants in the treatment group receive doses of vitamin D and participants in the control group receive a placebo. Neither the participants nor the researchers working directly with those participants know who is getting the vitamin D and who is getting the placebo, so we call it a double-blind procedure. After the data have been collected, the researcher will need to compare the memory scores for the two groups to see if the treatment worked. In all likelihood, the average test scores of the two groups will differ simply because they include two different groups of people. So how does the researcher know whether the difference is sufficient to conclude that vitamin D had an effect? This is where statistics come into play. Using certain statistical procedures, the researcher can state with an assured level of certainty (for example, with 95% confidence) that the disparity in average scores resulted from the vitamin D treatment. In other words, there is a slight possibility (in this case, 5%) that the difference was merely due to chance.

Thus, using statistical methods, researchers can determine whether their data reveal anything meaningful. In other words, they can establish statistical significance, which indicates that differences between groups in a study (for example, average scores for treatment and control groups) are so great that they are likely due to the researcher’s manipulations. If a research finding is statistically significant, the mathematical analyses have indicated that the probability those findings were due to chance is very minimal. When we use the experimental method (that is, randomly assign individuals, manipulate an independent variable, and control extraneous variables) and find significant differences between our experimental and control groups, we can be assured that the differences we see between these groups are very likely due to our manipulation (for example, vitamin D treatment versus a placebo) and not due to chance.

In addition to determining statistical significance, we also have to consider the practical importance of findings, which indicates the degree to which the findings of a study can be used in a meaningful way. In other words, do the findings have any relevance to real life? If our vitamin D regimen turns out to be statistically significant (with the performance gap between the treatment group and the control group most likely not due to chance), we still have to determine the practical importance of this disparity. If the two groups differ by only a few points on the cognitive test, we must ask ourselves if vitamin D supplementation is really worth the trouble.

Researchers may end up with differences between groups due to the size of their samples. Big samples are more likely to result in statistically significant results (small differences between groups can be amplified in some sense by the large sample) even though the results might not provide much practical information. The opposite can occur as well: A small sample size might hide an important difference between groups, which would only be apparent in a larger sample. Researchers can determine how big a sample is needed to detect differences between groups using statistical techniques (which you can learn more about in an elementary statistics class).

Sampling Techniques

Long before data are collected and analyzed, researchers must select people to participate in their studies. Depending on what a psychologist is interested in studying, the probability of being able to include all members of a population is not likely, so generally a sample, or subset of the population is chosen. The characteristics of the sample members must closely reflect those of the population of interest so that the researcher can generalize, or apply, her findings to the population at large.

One method for trying to ensure that the sample accurately reflects the larger population is to use random sampling, which means that any member of the population has an equally likely chance of being selected to participate in the study. A simple technique that can be used with a small population is to draw names or numbers from a hat: Every number in the hat has an equally likely chance of being drawn. If a researcher has a numbered list of the population members, she could generate random numbers on a computer and then contact the individuals with those numbers. Because the numbers are randomly picked, everyone on the list has an equal chance of being selected. Another approach is stratified sampling. With this method, the researcher decides that she wants a certain variable well represented—car ownership at the college, for example. She divides the population into four groups or strata (no car, one car, two cars, more than two cars), and then picks randomly from within each group or stratum. That way, she can be assured that the different types of car ownership are included in the sample. Often, researchers will use strata such as ethnicity, gender, and age groups to ensure that a sample has accurate proportions representing these important factors (such as an equal proportion of young adults, middle-aged adults, and elderly adults).

Some researchers use a method called convenience sampling, which entails choosing a sample from a group that is readily available or convenient. If student researcher is interested in collecting data on coffee drinking behavior from people who frequent coffee shops, he might be tempted to go to the Starbucks and Peet’s Coffee shops in his neighborhood. This approach may be better than nothing, but it certainly cannot be considered a random sample of coffee shop patrons. Just think of all the Dunkin’ Donuts and Caribou coffee drinkers who would be excluded from the sample.

Why does it matter if a convenience sample fails to randomly select participants? If it is not random, then the likelihood that it is a representative sample is very slim. In other words, a randomly picked sample is more likely than a convenience sample to include members with characteristics similar to the larger population. This is important because the whole point of conducting the study is for the researcher to be able to make inferences about the population based on the findings. Only if the sample is representative can the researcher make valid generalizations about the characteristics of the population.

It’s important to note that a randomly selected sample is not foolproof. There is always the possibility that the chosen group has characteristics that are not typical for the population. If this is the case, the sample is not truly representative, and our inferences might be incorrect. The smaller the sample, the less likely it will be representative, and the less reliable our findings. This is why larger samples are so desirable; they are more likely to provide accurate reflections of the population being studied.

How does this discussion of representative samples relate to statistics? Remember, the ultimate goal of the researcher is to use the results of a study to make inferences about the population at large. Every population is defined by various parameters, or numbers that describe its characteristics (for example, the average number of cars owned by sophomores in the United States). When the same characteristics are determined from a sample, they are referred to as statistics (the average number of cars owned by sophomores in the chosen random sample). (Note that the word “statistics” can also refer to the scientific discipline of collecting, organizing, analyzing, displaying, and interpreting data.) We will introduce you to these numerical characteristics later when we discuss measures of central tendency and measures of variation, but it is important to realize that without a representative sample, the use of statistics to make inferences about parameters is suspect. This is why it’s important to pay careful attention to the sample descriptions in media reports on scientific studies: Do they describe the samples, and are they truly representative? If not, the findings might only be true for the sample, not the population.

Variables

Once a study sample is selected, researchers can begin measuring and manipulating the variables of interest. Variables are measurable characteristics that vary over time or across people, situations, or objects. In psychology, variables may include cognitive abilities, social behaviors, or even the font size in books. Statisticians often refer to two types of variables. Quantitative variables are numerical, meaning they have values that can be represented by numbered units or ranks. Midterm exam scores, age at graduation, and number of students in a class are all quantitative variables. Qualitative variables are characteristics that are not easily quantified, but enable us to place members in categories. An example might be college major; you can ask all of the students in the library to line up under signs for psychology, biology, chemistry, undeclared, and so on, and thereby categorize them, but without a numerical rank. Of course we can rank how much we like the majors based on the courses associated with them, but they cannot be ordered or ranked in and of themselves. We can alphabetize them, but that is a ranking based on their labels. We can order the majors in terms of how many students are pursuing them, but that is a different variable (number of students). Other examples of qualitative variables include gender, ethnicity, and religious faith.

Variables are the focal point of a psychological experiment. Typically, the goal is to determine how one variable (the dependent variable) is affected by changes in another (the independent variable). Many studies focus on similar topics, so you might imagine it’s easy to compare their results. But this is not necessarily the case. Sometimes psychologists define variables in different ways, or study the same variables with vastly different samples, methods of measurement, and experimental designs. How do we reconcile all their findings? We rely on a meta-analysis, a statistical approach that allows researchers to combine the findings of different studies and draw general conclusions. A meta-analysis is an objective, quantitative (measurable) mechanism for gathering and analyzing findings from a set of studies on the same topic (Weathington, Cunningham, & Pittenger, 2010).