degree to which variables are related to one another. The most common measure of association is the Pearson product–moment correlation (symbolized as r). This statistic describes how strongly (or weakly) variables are related to one another. For example, if two variables are perfectly correlated, the r value will be 1 or –1. The sign of the number indicates the direction of the relationship. A positive correlation tells us that the variables are directly related; as one variable increases, so does the other, and as one variable decreases, so does the other. A negative correlation tells us that the variables are inversely related. That is, as one variable increases, the other decreases, and as one variable decreases, the other increases. The magnitude of r tells us how strongly the variables are related. A zero correlation tells us that the variables are not related at all; as the value increases to +1 or decreases to –1, the strength of the relationship increases. A correlation of 1 (either positive or negative) is called a perfect correlation.Be aware that perfect correlations never actually occur in the real world. If they do, it usually means that you have inadvertently measured the same variable twice and correlated the data. For example, you would likely get a correlation of 1 if you measured reaction time in seconds and also in minutes. It would be no surprise to find that the values are correlated because they are the same measure, only in different scales. Here is another example: Suppose you measured mood with two scales. It is likely that the measures will correlate highly. Again, this only indicates that you have two measures of the same thing.
These descriptive statistics are used to summarize what was observed in the research. But the idea of a lot of research is to generalize the findings beyond just the observations or participants in the study. We ultimately want to say something about behavior in general, not just the behavior that occurred in the study. To make these generalizations, we need inferential statistics. Before leaping into a list of the various inferential statistics you will likely come across in the literature, we would like to review some of the basic concepts of inference.
Inferential Statistics
Inferential statistics are used to generalize the findings of a study to a whole population. An inference is a general statement based on limited data. Statistics are used to attach a probability estimate to that statement. For example, a typical weather forecast does not tell you that it will rain tomorrow afternoon. Instead, the report will indicate the probability of rain tomorrow. Indeed, the forecast here for tomorrow is a 60% chance of rain. The problem with making an inference is that we might be wrong. No one can predict the future, but based on good meteorological information, an expert is able to estimate the probability of rain tomorrow. Similarly, in research, we cannot make generalized statements about everyone when we only include a sample of the population in our study. Instead, we attach a probability estimate to our statements.
When you read the results of research articles, the two most common uses of inferential statistics will be hypothesis testing and confidence interval estimation.
Does wearing earplugs improve test performance?
Is exercise an effective treatment for depression?
Is there a relationship between hours of sleep and ability to concentrate?
Are married couples happier than single individuals?
These are all examples of research hypotheses that could be tested using inferential tests of significance. What about the following?
Does the general public have confidence in its nation’s leader?
How many hours of sleep do most adults get?
At what age do most people begin dating?
These are all examples of research with a focus on describing attitudes and/or behavior of a population. This type of research, which is more common in sociology than in psychology, uses confidence interval estimation instead of tests of significance.
The vast majority of psychological research involves testing a research hypothesis. So let’s first look at the types of tests of significance you will likely see in the literature and then look at confidence intervals.
Common Tests of Significance.
Results will be referred to as either statistically significant or not statistically significant. What does this mean? In hypothesis-testing research, a straw person argument is set up where we assume that a null hypothesis is true, and then we use the data to disprove the null and thus support our research hypothesis. Statistical significance means that it is unlikely that the null hypothesis is true, given the data that were collected. Nowhere in the research article will you see a statement of the null hypothesis; instead, you will see statements about how the research hypothesis was supported or not supported. These statements will look like this:
With an alpha of .01, those wearing earplugs performed statistically significantly better (M = 35, SD = 1.32) than those who were not (M = 27, SD = 1.55), t(84) = 16.83, p = .002.
The small difference in happiness between married (M = 231, SD = 9.34) and single individuals (M = 240, SD = 8.14) was not statistically significant, t(234) = 1.23, p = .21.
These statements appear in the results section and describe the means and standard deviations of the groups and then a statistical test of significance (t test in both examples). In both statements, statistical significance is indicated by the italic p. This value is the p value. It is an estimate of the probability that the null hypothesis is true. Because the null hypothesis is the opposite of the research hypothesis, we want this value to be low. The accepted convention is a p value lower than .05 or, better still, lower than .01. The results will support the research hypothesis when the p value is lower than .05 or .01. The results will not support the research hypothesis when the p value is greater than .05. You may see a nonsignificant result reported as ns with no p value included.
You will find a refresher on statistical inference, including a discussion of Type I and Type II errors, and statistical power in Chapter 4.
Researchers using inferential techniques draw inferences based on the outcome of a statistical significance test. There are numerous tests of significance, each appropriate to a particular research question and the measures used, as you will recall from your introductory statistics course. It is beyond the scope of our book to describe in detail all or even most of these tests. You might want to refresh your memory by perusing your statistics text, which of course you have kept, haven’t you? We offer a brief review of some of the most common tests of significance used by researchers in the “Basic Statistical Procedures” section of this chapter.
Going back to the results section of our example article, we see that the author has divided that section into a number of subsections. The first subsection, with the heading “Mood,” reports the effect of light on mood. It is only one sentence: “No significant results were obtained” (Knez, 2001, p. 204). The results section is typically brief, but the author could have provided the group means and the statistical tests that were not statistically significant. The next subsection, titled “Perceived Room Light Evaluation,” provides a statistically significant effect. Knez (2001) reports a significant (meaning statistically significant) gender difference. He reports Wilks’ lambda, which is a statistic used in multivariate ANOVA (MANOVA; when there is more than one DV), and the associated F statistic and p value for the gender difference, F (7, 96) = 3.21, p
