A variety of statistical procedures are available for answering such questions. Which procedure will be appropriate in a given case depends on the kind of data found in the variables being compared. Variables appear in two principal forms: (a) as a distribution of scores or measurements ranging from low to high, (b) as a series of rankings. Table 11-1 indicates which correlation process is suited to which pattern of data. Each of the correlation procedures listed in the right-hand column of Table 11-1 is described briefly in the following pages.

Table 11-1
Kinds of Data and Correlation Procedures

Pearson product-moment correlation. The research question: What is the degree of relationship between two variables if each variable consists of measurements along a scale that consists of a series of equal intervals?

In judging whether the Pearson product-moment method (symbolized by the letter r) is appropriate for a given study, the researcher needs to decide whether both of the variables being compared represent equal-interval scales. Strictly speaking, an equal-interval scale is one in which the distance from one step to the next step is precisely the same throughout the entire length of the scale. Measures of height fulfill this equal-interval requirement, since the distance from 25 centimeters to 30 centimeters is exactly the same as from 60cm to 65cm or from 105cm to 110cm, and so on. Measures of temperature, of weight, of time, of speed, of dollars, and of distance on a running track also involve equal-unit scales. In contrast, some scales used in the social sciences may appear at first glance to be composed of equal intervals, but upon closer inspection it becomes clear that they are not. The scoring of intelligence test results in terms of IQ levels is a familiar example. The difference in intellectual ability between an IQ of 100 and one of 105 is not the same as the difference between 120 and 125, because the questions that make up the test are not all of equal difficulty. Test items that differentiate between IQs 120 and 125 are probably more demanding than those that distinguish between IQs 100 and 105. Likewise, a distribution of scores on a test of English usage or of science facts does not produce an equalinterval scale in the strict meaning of the term, because the items that differentiate between scores 80 and 90 are likely more difficult than those that differentiate between scores of 30 and 40. In effect, a 10-point difference in one segment of the scale is not equal to a 10-point difference in another segment.

This distinction between truly equal-interval scales and scales that contain intervals that are only approximately equal has caused some critics to condemn the use of the Pearson method in a great many of the studies that have employed the procedure. (More than 90% of all correlation coefficients reported in the research literature within the behavioral sciences are Pearsonr's.) However, Heermann and Braskamp ( 1970, pp. 30-110) analysis of a host of investigations suggests that using Pearson's technique with variables involving intervals that are no more than approximately or partially equal is still warranted. Glass and Hopkins ( 1984, p. 9) have observed that the critics' "disenchantment with the classical methods was premature." Thus, it is generally acceptable to apply Pearson's r in studies whose variables consist of test scores or involve ratings of performance or of attitude.

Spearman rank correlation. The research question: What is the degree of relationship between two variables if each variable consists of ranks along a scale rather than measured intervals?

Frequently research data are not in the form needed to compute a Pearsonr, because one or both of the variables consist of ranks instead of measured amounts. Sometimes data are originally collected as rankings, as when teachers are ranked in terms of popularity with students, basketball players ranked by overall ability, and nations ranked by the prestige of their higher-education systems. Other times the data are collected as quantities which are then converted into ranks for convenience of computation. Thus, provinces can be ranked by per-pupil expenditures, colleges by their graduation rates, and students by their grade point averages. Probably the most popular rank-order correlation statistic is Spearman's rho (ρ).

Biserial correlation. The research question: What is the degree of relationship between two variables if one is measured in a graduated fashion so as to produce a sequence of quantities and the other variable is in the form of a dichotomy?

Two computational techniques for determining the association between such variables are the biserial and point-biserial methods. Each yields correlation coefficients that are estimates of what the Pearson r would be if both variables were normally distributed arrays rather than one of them being in the form of a dichotomy.

Deciding whether the biserial technique is appropriate in a given research situation depends on the researcher's assumption about the nature of the dichotomous variable. The biserial method is not appropriate in cases of true dichotomies, such as sex (male/female) or employees' attendance at work on a particular day (present/absent). However, it is applicable when the dichotomy appears to be an artifact of crude measurement. For instance, in a survey of parents' opinions about teaching birth control methods in high school, data may be collected in the form of a dichotomy (agree/disagree). But it is likely that parents' opinions are actually far more varied than the resulting data suggest-- some parents will have strong objections to birth control instruction, some will disagree moderately, others will object mildly, some will agree but with serious reservations, and so on. If a more precise scaling approach had been used in gathering opinions, the results would have assumed the form of a distribution of graduated steps. Thus, the dichotomized variable in this instance was not truly discrete. Therefore, it is this latter type of spurious, crude-measurement dichotomy for which the biserial correlation technique is designed.

On the other hand, if the dichotomous feature is truly discrete (male/female, citizen/alien, fourth-grade pupil/non-fourth-grade pupil), an estimate of r can still be obtained by applying the point-biserial method.

Phi coefficient. The research question: What is the degree of relationship between two variables if both of them consist of sets of discrete categories?

The phi (φ) coefficient is the product-moment correlation between two variables when each variable is scored as discrete points rather than as a series of measured steps. For example, imagine that we wish to determine among political party activists the direction and degree of relationship on a given work day between two variables: (a) the activists' marital status and (b) promptness. We identify two types of marital status (married and single) and two levels of promptness in arriving at work (on-time and late). We then construct a two-bytwo table with the marital-status variable on the horizontal axis and promptness variable on the vertical axis. We can now enter data about the relationship between an individual's marital condition and promptness into the four cells of our two-by-two table. This then permits us to computer a phi coefficient reflecting the degree of relationship between our two variables.

The data used in computing phi coefficients need not be restricted to two discrete positions on each of the variables. For instance, comparing college students' class levels (frosh, soph, junior, senior) and those students' use of alcohol during a given week (did drink vs. did not drink) would produce a 4-by-2 table.

In like manner, other variables represented in discrete types or steps could produce larger size tables--4-by-4, 3-by-6, and such.

Other correlation options. The correlation methods described in the above paragraphs are only four of the more commonly used techniques. Numerous other approaches found in statistics textbooks and journal articles are designed to suit additional conditions of the data that a researcher has at hand. For instance, in some situations the relationship between two series of measures may not assume the shape of a straight diagonal line. As scores on one variable increase, the scores on the other do not increase regularly in a similar manner. Such a relationship results if people's ages over their life span are compared with their eyehand coordination scores. Whereas age increases in regular steps, eye-hand skills do not; instead, such skills increase in early life, remain at a high level for much of adulthood, then decline in old age, thereby rendering their progression curvilinear. In such cases, an eta (η) coefficient can be computed to reflect the association between the variables. As a second example, under certain conditions a tetrachoric coefficient (r _t ) rather than a phi coefficient can usefully be calculated to determine the magnitude of the relationship between two variables, each of which is a dichotomy.

The term factor analysis identifies several alternative procedures for estimating which features are common to a series of correlations that have been computed from a variety of measures of a group of individuals. For example, a large number of students can be administered tests intended to assess their mental abilities. Correlations can then be computed to determine which test items are highly related to each other and which ones appear to be mainly independent of each other. The assumption is that when certain items are closely associated (so that students who do well on one item in the cluster also do well on the others, and vice versa), a particular mental ability or mental factor underlies that group of items. Typically, a label is assigned to that cluster of closely related items, with the label intended to reflect the cognitive skill--or factor--that binds the cluster together. For example, the labels applied to factors found in such test batteries as the Primary Mental Abilities ( Thurstone, 1938) and Differential Aptitude Tests ( Bennett, Seashore, & Wesman, 1952) are number comprehension, verbal reasoning, verbal comprehension, abstract reasoning, clerical speed and accuracy, mechanical reasoning, space relations, language usage, and word fluency.

In applications in education, factor analytic studies have been undertaken in such diverse areas as prose style, administrative behavior, occupational classification, attitudes and belief systems, and the economics of education. The technique is still in extensive use in the exploration of abilities, in the refining of tests and scales, and in the development of composite variables for use in research studies. Its most promising applications in recent years, however, have been concerned with the testing of explicit hypotheses about the structure of sets of variables, as in the study of growth models. . . . It has also facilitated the comparison of the factorial structure of different subpopulations, allowing investigators to determine whether the factorial structure of a given set of variables varies, for example, with sex, age, ethnicity, socioeconomic status, or political affiliation. ( Spearritt, 1985, pp. 1822-1823)

Drawing Inferences from Samples

As mentioned earlier, descriptive statistics summarize in a concise form the results of measurements of a group of individuals or events. Sometimes researchers are interested only in what such statistics tell about that group. However, other times they want to apply the group's results to a larger population. In other words, as described in Chapter 7, the measured group is considered to be a sample of a larger population that has not been measured. Hence, from testing the reading ability of 200 nine-year-olds, an investigator may intend to draw inferences about the reading skills of all of a city's or state's nine-year-olds. From a statistical summary of 350 religious workers' expressed attitudes about the use of marijuana, a researcher may hope to estimate the attitudes toward illicit drugs of all such religious workers. However, extending the conclusions about a tested group to a larger population always entails a risk of error, since the sample group may not truly represent the larger population. In effect, the sample may be biased. Therefore, it's important for researchers to have ways of judging how likely the statistics gathered about a sample will accurately portray the features of an intended population. Or, stated as a question, what is the probability of making an error when using descriptive statistics as the basis for drawing inferences about a population? The procedures for answering such a question are called inferential statistics.

It's useful at this point to consider the sources of errors that may distort the conclusions drawn from assessing people or events. In the case of descriptive statistics, inaccurate conclusions derive from measurement errors. For instance, the purpose of having students take a history test is to discover precisely their knowledge of historical facts, concepts, trends, theories, and the like. However, various kinds of errors can render the assessment inaccurate. The directions for taking the test may be unclear, some test items may be badly phrased, noises in the classroom may disrupt students' attempt to concentrate, the time to complete the test may be too short, the tester's method of correcting the students' answers may be faulty, and more. Such measurement errors can be reduced by careful attention to the preparation of the test, to the manner of administering it, and to the method of correcting it. However, if the results of testing a sample of students are used as the foundation for drawing inferences about the broader population of students from which the sample was drawn, another source of inaccuracy can distort the inferred picture of the population's knowledge of history. That source is sampling error, meaning the degree to which inferences about a population likely deviate from the true characteristics of that population. The following discussion identifies two popular statistical procedures for estimating the magnitude of sampling error.

As noted, researchers can never know for sure how accurately a sample drawn from a population reflects the characteristics of that population. For instance, assume that you conduct telephone interviews with 100 consumers to learn which TV programs they prefer, and you compute the percentages of your respondents who prefer various programs. What you now want to know is how likely those percentages are an accurate reflection of the preferences of all 500,000 residents of the city in which you are conducting your survey. The only way you can know for sure the accuracy of your results is to interview all 500,000. But since interviewing the entire population would be impractical, the best you can do is to estimate the probability that the sample percentages are close to the population's true percentages. Inferential statistics are designed to furnish that estimate. We will briefly inspect two of the ways to arrive at such estimates--the t-test and the analysis of variance.

The t-test

Researchers often compare two groups in terms of their means. If the means are found to differ, the question arises: Does each group represent a different population in relation to the characteristic that was measured, so the difference in these sample means reflects an actual difference in the means of the underlying populations? Or are the two groups simply two slightly biased samples from the same population, whose true mean we really don't know? To illustrate, imagine that 50 women and 50 men are enrolled in a college class entitled "Methods of Logic." On the final test at the end of the semester, the mean for the women is 83.6 and for the men 78.9. We may now ask whether these scores reflect a difference only between female and male members of that particular class, or is the population of the kind of college women who enrolled in the class generally more adept at learning the methods of logic taught in the class than is the population of the kind of college men who enrolled? The t-test provides an estimated answer to this query.

By applying the appropriate computation procedure (found in nearly any statistics textbook), we learn that there apparently is less than 1 chance in 1000 that the two groups represent the same population and that the obtained means are different simply because of bias in drawing the samples. In other words, our results support the conclusion that the population of women (of the kind enrolled in the logic class) is on the average somewhat more skilled at learning the methods of logic taught in the class than is the population of men (of the kind that enrolled). There is a 999 chance in 1,000 that this conclusion is warranted.

However, if the means for women in our hypothetical logic class had been 81.0 and the men 83.6 (with the standard deviations σ = 6.7 and σ = 6.3), we would learn that there are likely 5 chances out of 100 that there is no real difference in the means of the populations from which these women and men were drawn. In effect, there are 5 chances in 100 that the difference between 81.0 and 83.6 is simply the result of sampling error--the men's sample just happened to include more adept logic learners--and that both the men and women represent the same population in terms of ability to master the logic techniques taught in the class. But there are 95 chances in 100 that the obtained differences actually do reflect a difference that would be found in the mean scores of the two populations of the kinds of women and men who took the test.

Thus, the t-test is designed to help researchers estimate the probability that measures of a sample of people or events accurately portray the broader populations of people or events from which the sample was apparently drawn. In addition to testing the representativeness of obtained means, there are t-tests for pairs of medians, percentages, standard deviations, and correlations.

In the above brief sketch of the t-test procedure, we have not taken the space to point out several important assumptions about the way samples are drawn from populations, assumptions that significantly affect the appropriateness of t-tests in particular studies. For explanations of those assumptions, readers are directed to the suggested readings at the end of this section.

Analysis of variance

As explained earlier, the variance (σ²) is a description of how much measurements spread away from the center of a distribution. Specifically, the variance is the average of the squared measurement deviations from the mean.

We have seen that the t-test is used to estimate whether the means from two samples represent the same population or two different populations. The analysis of variance (ANOVA) is a procedure for simultaneously testing how likely three or more means represent samples drawn from the same population or, in contrast, are means representing different populations. One example of comparing three or more means is found in attitudes toward a birth control methods as expressed by parents, teachers, police officers, and teenagers. Another example is found in a study of mathematics test scores of high school students representing six ethnic groups--Anglos, Latinos, Afro-Americans, Asians, Native Americans, and Pacific Islanders.

Not only does ANOVA permit the simultaneous comparison of multiple means, but the results are more accurate than if t-tests were applied to each pairing of the multiple means being studied. Glass and Hopkins ( 1984, p. 324) point out that "ANOVA is the most common of all inferential statistical techniques in education and the behavioral sciences."

ANOVA results are interpreted in much the same way as those of the t-test, that is, in terms of the probability that a difference between sample means are the result of sampling error rather than the result of a difference in the true means of the populations from which the compared samples were drawn. Thus, a difference among sample means that could occur by chance (by sampling error) at a probability level of only 1 time in 100 gives the researcher more confidence in believing that the means of the represented populations are truly different than does a difference among sample means that could occur by chance 5 times in 100 or 10 times in 100.ANOVA can also be extended to test the likelihood of interactions among factors. For instance, one researcher used ANOVA to discover whether teachers' ethnic status affected their perceptions of how adaptable Anglo and Latino students were. The results showed that there was indeed interaction between teacher and student ethnic types. Latino teachers more often judged Latino students as more adaptive, whereas Anglo teachers more frequently considered Anglo students more adaptive ( Glass & Hopkins, 1984, p. 404).

Additional options

In the foregoing pages we have introduced a few types of statistics commonly used in research and have suggested which of the types are most useful for answering various kinds of research questions. There are, however, far more statistical procedures than those reviewed in this chapter, procedures well suited to answering additional kinds of research questions. Such additional types of statistical treatments are identified by such titles as: chi square, partial correlation, one-way analysis of variance, two-way analysis of variance, the analysis of covariance, linear and nonlinear regression, Kendall's tau coefficient, Kendall's coefficient of concordance, the median test, the Mann-Whitney U test, the WoldWolfowitz runs test, and others.Descriptions of a wide range of statistical procedures, as well as their computational steps, are found in such sources as the following.

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