All the NHSTs in previous chapters compare two dependent variable means. When there are three or more group means, it is possible to use unpaired two-sample t-tests for each pair of group means, but there are two problems with this strategy. First, as the number of groups increases, the number of t-tests required increases faster. Second, the risk of Type I error increases with each additional t-test.

The analysis of variance (ANOVA) fixes both problems. Its null hypothesis is that all group means are equal. ANOVA follows the same eight steps as other NHST procedures. ANOVA produces an effect size, η2. The η2 effect size can be interpreted in two ways. First, η2 quantifies the percentage of dependent variable variance that is shared with the independent variable’s variance. Second, η2 measures how much better the group mean functions as a predicted score when compared to the grand mean.

ANOVA only says whether a difference exists – not which means differ from other means. To determine this, a post hoc test is frequently performed. The most common procedure is Tukey’s test. This helps researchers identify the location of the difference(s).

In this chapter, there are two types of probabilities that can be estimated: empirical probability and theoretical probability. Empirical probability is calculated by conducting a number of trials and finding the proportion that resulted in each outcome. Theoretical probability is calculated by dividing the number of methods of obtaining an outcome by the total number of possible outcomes. Adding together the probabilities of two different events will produce the probability that either one will occur. Multiplying the probabilities of two events together will produce the probability that both will occur at the same time or in succession. As the number of trials increases, the empirical probability and theoretical probability converge.

It is possible to build a histogram of empirical or theoretical probabilities. As the number of trials increases, the empirical and theoretical probability distributions converge. If an outcome is produced by adding together (or averaging) the results of events, the probability distribution is normally distributed. Because of this, it is possible to make inferences about the population based on sample data – a process called generalization. The mean of sample means converges to the population mean, and the standard deviation of means (the standard error) converges on the value.

This chapter discusses two types of descriptive statistics: models of central tendency and models of variability. Models of central tendency describe the location of the middle of the distribution, and models of variability describe the degree that scores are spread out from one another. There are four models of central tendency in this chapter. Listed in ascending order of the complexity of their calculations, these are the mode, median, mean, and trimmed mean. There are also four principal models of variability discussed in this chapter: the range, interquartile range, standard deviation, and variance. For the latter two statistics, students are shown three possible formulas (sample standard deviation and variance, population standard deviation and variance, and population standard deviation and variance estimated from sample data), along with an explanation of when it is appropriate to use each formula. No statistical model of central tendency or variability tells you everything you may need to know about your data. Only by using multiple models in conjunction with each other can you have a thorough understanding of your data.

Pearson’s correlation describes the relationship between two interval- or ratio-level variables. Positive correlation values indicate that individuals who have high X scores tend to have high Y scores (and that individuals with low X scores tend to have low Y scores). A negative correlation indicates that individuals with high X scores tend to have low Y scores (and that individuals with low X scores tend to have high Y scores). Correlation values closer to +1 or –1 indicate stronger relationships between the variables; values close to zero indicate weaker relationships. A correlation between two variables does not imply a causal relationship between them.

It is also possible to test a correlation coefficient for statistical significance, where the null hypothesis is r = 0. This follows the same steps of all NHSTs. The effect size for Pearson’s r is calculated by squaring the r value (r2).

A correlation is visualized with a scatterplot. Scatterplots for strong correlations have dots that are closely grouped together; scatterplots showing weak correlations have widely spaced dots. Positive correlations have dots that cluster in the lower-left and upper-right quadrants of a scatterplot. Negative correlations have the reverse pattern.

One of the most frequent research designs in the social sciences is to compare two groups’ scores. When there is no pairing between sets of scores, it is necessary to conduct an unpaired two-sample t-test. The steps of this NHST are the same eight steps of the previous statistical tests because all of these are members of the GLM. There are some modifications that make the unpaired two-sample t-test unique:

• The null hypothesis for an unpaired two-sample t-test is always.

• The equation for the number of degrees of freedom is different: df = (n1 – 1) + (n2 – 1).

• The formula to calculate the observed value is more complex:.

• The effect size for an unpaired two-sample t-test is still Cohen’s d, but the formula has been modified:.

The unpaired two-sample t-test shares many characteristics with other NHSTs: the sensitivity to sample size, the necessity of calculating an effect size, the arbitrary nature of selecting an α value, and the need to worry about Type I and Type II errors.

Paired scores occur when each score in one sample corresponds to a single score in another sample. Whenever paired scores occur, researchers can measure whether the difference between them is statistically significant through a paired-samples t-test. A paired-samples t-test is similar to a one-sample t-test. The major difference is that the sampling distribution consists of differences between group means. This necessitates a few minor changes to the t-test procedure. The major differences are:

• The null hypothesis for a paired-samples t-test is now, whereis the average difference between paired scores.

• n now refers to the number of score pairs in the data.

• The Cohen’s d formula is now, where the numerator is the difference between the two sample means and is the pooled standard deviation.

All of the other aspects of a paired-samples t-test are identical to a one-sample t-test, including the default α value, the rationale for selecting a one- or a two-tailed test, how to compare the observed and critical values in order to determine whether to reject the null hypothesis, and the interpretation of the effect size. The caveats of all NHSTs apply.