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appendix a

introduction to statistics

The vast knowledge base that defines the field of psychology is the result of rigorous and meticulous scientific research, most of which entails the careful collection of data. In Chapter 1, we presented various methods used to gather this data, but we only touched upon statistical approaches for analyzing it. Here, we will discover how we can use data meaningfully: Welcome to statistics, the science of collecting, organizing, analyzing, displaying, and interpreting data.

Statistics are everywhere—not just in the academic materials published by psychologists. Newspapers, Web sites, and television shows report on statistical findings every day, though they sometimes make mistakes, exaggerate, or leave out important information. You can detect these types of errors if you understand statistics. It is important for everyone (not just psychology students) to think critically about research findings.

There are two basic types of statistics: descriptive and inferential. With descriptive statistics, researchers summarize information they have gleaned from their studies. The raw data can be organized and presented through tables, graphs, and charts, examples of which we provide 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 we will revisit later. The goal of descriptive statistics is to describe data, or provide a snapshot of what is observed in a study. Inferential statistics, on the other hand, go beyond simple data presentation. With inferential statistics, for example, we can determine the probability of events and make predictions about general trends. The goals are 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. Hypothesis testing involves mathematical procedures to determine whether data support a 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? Using statistical procedures, the researcher can state with a chosen 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.

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With the use of statistical methods, researchers can establish statistical significance, indicating that differences between groups in a study (for example, the average scores for treatment and control groups) are so great that they are likely due to the researcher’s manipulations; the mathematical analyses suggest a minimal probability the findings were due to chance. When we use the experimental method (that is, randomly assign individuals, manipulate an independent variable, and control extraneous variables) and find statistically significant differences between our experimental and control groups, we can be assured that these differences are very likely due to how we treated the participants (for example, administering vitamin D treatment versus a placebo).

In addition to determining statistical significance, we also have to consider the practical importance of findings, meaning the degree to which the results of a study can be used in a meaningful way. In other words, do the findings have any relevance to real life? If the vitamin D regimen produces statistically significant results (with a performance gap between the treatment and control groups most likely not due to chance), the researcher would still have to determine its practical importance. Suppose the two groups differ by only a few points on the cognitive test; then the question is whether vitamin D supplementation is really worth the trouble. We should note that big samples are more likely to result in statistically significant results (small differences between groups can be amplified by a large sample) even though the results might not provide much practical information.

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.

In an effort to ensure that the sample accurately reflects the larger population, a researcher may use random sampling, which means that all members of the population have an equal chance of being invited to participate in the study. If the 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. A researcher chooses this method if she wants a certain variable to be well represented—car ownership in urban areas, 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, ensuring that all of the different types of car ownership are included in the sample. Researchers use strata such as ethnicity, gender, and age group to ensure a sample has appropriate representation of these important factors.

Some researchers use a method called convenience sampling, which entails choosing a sample from a group that is readily available or convenient. If a 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. But this approach does not use random sampling (just think of all the Dunkin’ Donuts and Caribou coffee drinkers who would be excluded), so the likelihood that it results in 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 population. Only if a sample is representative can a researcher use his findings to make accurate inferences or valid generalizations about the characteristics of the population. But it’s important to note that even a randomly selected sample is not foolproof. There is always the possibility that the chosen participants have characteristics that are not typical for the population. The smaller the sample, the less likely it will be representative and the less reliable the results. Larger samples tend to provide more accurate reflections of the population being studied.

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The ultimate goal of most studies is to provide results that can be used to make inferences about a population. We can describe a population using various parameters, or numbers that delineate its characteristics (for example, the average number of cars owned by all households in urban areas in the United States). When the same characteristics are determined for a sample, they are referred to as statistics (the average number of cars owned by households in the sample). (Recall that the word “statistics” can also refer to the scientific discipline of collecting, organizing, analyzing, displaying, and interpreting data.) We will introduce you to some of these numerical characteristics later when we discuss measures of central tendency and measures of variation.

Understanding sampling techniques can help you become a more critical consumer of scientific information. When reading or watching media reports on scientific studies, ask yourself whether the samples are truly representative. If not, the use of statistics to make inferences about parameters is suspect; the findings might only be true for the sample, not the population.

Once a study sample is selected, researchers can begin studying 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 enable us to place participants in categories, but they cannot be assigned numbered units or ranks. 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 by their majors. We can rank how much we like the majors based on the courses associated with them, but the majors cannot be ordered or ranked in and of themselves. We can alphabetize them, but that is a ranking based on their labels. We can even 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.

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Variables are the focal point of experiments in psychology. 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).