Samuel T. Mayo

Simple Analysis of Variance

Back in Chapter 5, we learned to use the t test to determine whether or not two means differ significantly from each other. In this chapter, we will learn to use analysis of variance to determine whether or not several means differ significantly from each other.

Interactions Among Categorical Variables

OTHER papers have considered interactions among continuous, or nicely scaled, variables. Here the concern is with categorical variables, or attributes. The terms will be used interchangeably. The term &dquo;attribute&dquo; will be defined as a set of mutually exclusive and exhaustive &dquo;categories,&dquo; so that any person in a sample can be classified as belonging in one and only one category of a given attribute. The simplest set is the dichotomy, in which the observer merely notes the presence or absence of the attribute and in which the categories are &dquo;has the attribute&dquo; versus &dquo;does not have the attribute.&dquo; In other sets of more than two categories, one can have ordered categories or purely qualitative, or nominal, ones. Interaction between two categorical variables as tested by chisquare in a two-way contingency table is familar to all of us. It is

Skewness and Transformed Scores

We have already noticed that many frequency distributions are symmetrical, or nearly so, while some (such as the two shown in Figure 7.1) are markedly lopsided or skewed. Skewness is defined as the extent to which a frequency distribution departs from a symmetrical shape.

More Measures of Correlation

Under most circumstances, if it is physically possible to compute it, the Pearson product moment coefficient of correlation is the best measure of relationship and should be used. There are, however, at least twenty other measures of correlation. Except for the measurement of curvilinear relationship, all the others are merely imitations of the Pearson r we studied in Chapter 8.

Nonparametric Statistics Other than Chi Square

Prior to the preceding chapter on chi square, the great majority of the statistics we computed were parametric statistics but there was no occasion to use that expression in referring to them. These statistics frequently made assumptions (such as normality) about the nature of the population from which the samples were drawn. Because the population values are called parameters, such statistics can be called parametric, although the word is not often used.

The Nature of Statistical Methods

Numbers appear in newspapers every day. Corporations assemble large quantities of data for their own use. Federal and state governments collect and summarize thousands upon thousands of pages of data. Scientists record the results of their observations. Many other individuals gather information in numerical form. These figures are not ends in themselves, but when properly analyzed and interpreted, they form sound bases for important decisions, valid conclusions, and appropriate courses of action.

Pearson Product Moment Coefficient of Correlation

In the first seven chapters of this text, we have dealt with situations in which each one of the N people in our study had a score, X, on some test or other measurement. It is now time for us to expand our view and consider the perfectly realistic situation in which each individual may have scores on two (or more) different variables. Suppose that we have the height and weight of each one of a group of 12-year-old girls. We can, of course, compute the mean and standard deviation of height; and we can do the same for weight. But a more important question may well be: Is there a correlation, or relationship, between height and weight? Are the tall girls apt to weigh more or less than the short girls? Do the heavy girls tend to be above or below the average in height?

Normal Probability Curve

By far the best known distribution curve in statistics is the normal probability curve.* Unlike some curves with two peaks, this one has only one high point. One reason it is so well known is that it has so many uses. Since the normal probability curve is widely used, it is important for us to know about some of its uses.

Standard Errors of Differences

Because of our feeling that standard errors should be closely associated with the statistics to which they pertain, this book was so arranged as to make it practicable to place the chapter on Statistical Inference near the beginning of the text—after only a few statistics (X, Mdn, s, and AD) had been presented. Subsequently, as new statistical terms were introduced they were usually accompanied by their standard errors. From these standard errors, it was a very simple matter to determine confidence limits whenever they were needed. And from these limits we could find out whether or not our sample statistic differed significantly from any hypothesized population parameter. But such parameters were always fixed points, not figures subject to variation, as the sample statistics were.

Percentiles and Percentile Ranks

A percentile is a point below which a given percentage of the cases occur. For instance, 67 percent of the cases are below the sixty-seventh percentile, which is written as P 67. The middle of a distribution, or the point below which 50 percent of the cases lie, is, of course, the fiftieth percentile. In Chapter 2 we learned another name for this value—the median. The median is probably computed, used, and discussed more than any other percentile.