gt;gt; Chapter One gt; gt;A CONCEPTUAL FOUNDATIONgt; gt; gt; If you have not already read the Preface, please do so now. Many readers have developed the habit of skipping the Preface because it is often used by the author as a soapbox, or as an opportunity to give his or her autobiography and to thank many people the reader has never heard of. The preface of this text is different and plays a particularly important role. You may have noticed that this book uses a unique form of organization (each chapter is broken into A, B, and C sections). The preface explains the rationale for this unique format and explains how you can derive the most benefit from it. gt; gt; gt;What Is (Are) Statistics?gt; gt; An obvious way to begin a text about statistics is to pose the rhetorical question, "What gt;isgt; statistics?" However, it is also proper to pose the question "What gt;aregt; statistics?"—because the term gt;statisticsgt; can be used in at least two different ways. In one sense gt;statisticsgt; refers to a collection of numerical facts, such as a set of performance measures for a baseball team (e.g., batting averages of the players) or the results of the latest U.S. census (e.g., the average size of households in each state of the United States). So the answer is that statistics are observations organized into numerical form. gt; In a second sense, gt;statisticsgt; refers to a branch of mathematics that is concerned with methods for understanding and summarizing collections of numbers. So the answer to "What is statistics?" is that it is a set of methods for dealing with numerical facts. Psychologists, like other scientists, refer to numerical facts as gt;datagt;. The word gt;datagt; is a plural noun and always takes a plural verb, as in "the data gt;weregt; analyzed." (The singular form, datum, is rarely used.) Actually, there is a third meaning for the term gt;statisticsgt;, which distinguishes a statistic from a parameter. To explain this distinction, I have to contrast samples with populations, which I will do at the end of this section. gt; As a part of mathematics, statistics has a theoretical side that can get very abstract. This text, however, deals only with gt;applied statisticsgt;. It describes methods for data analysis that have been worked out by statisticians, but does not show how these methods were derived from more fundamental mathematical principles. For that part of the story, you would need to read a text on gt;theoreticalgt; or gt;mathematical statisticsgt; (e.g., Hogg & Craig, 1995). gt; The title of this text uses the phrase "psychological statistics." This could mean a collection of numerical facts about psychology (e.g., how large a percentage of the population claims to be happy), but as you have probably guessed, it actually refers to those statistical methods that are commonly applied to the analysis of psychological data. Indeed, just about every kind of statistical method has been used at one time or another to analyze some set of psychological data. The methods presented in this text are the ones usually taught in an intermediate (advanced undergraduate or graduate level) statistics course for psychology students, and they have been chosen because they are not only commonly used but are also simple to explain. Unfortunately, some methods that are now used frequently in psychological research (e.g., structural equation modeling) are too complex to be covered adequately at this level. gt; One part of applied statistics is concerned only with summarizing the set of data that a researcher has collected; this is called descriptive statistics. If all sixth graders in the United States take the same standardized exam, and you want a system for describing each student's standing with respect to the others, you need descriptive statistics. However, most psychological research involves relatively small groups of people from which inferences are drawn about the larger population; this branch of statistics is called gt;inferential statisticsgt;. If you have a random sample of 100 patients who have been taking a new antidepressant drug, and you want to make a general statement about the drug's possible effectiveness in the entire population, you need inferential statistics. This text begins with a presentation of several procedures that are commonly used to create descriptive statistics. Although such methods can be used just to describe data, it is quite common to use these descriptive statistics as the basis for inferential procedures. The bulk of the text is devoted to some of the most common procedures of inferential statistics. gt; gt; gt;Statistics and Researchgt; gt; The reason a course in statistics is nearly universally required for psychology students is that statistical methods play a critical role in most types of psychological research. However, not all forms of research rely on statistics. For instance, it was once believed that only humans make and use tools. Then chimpanzees were observed stripping leaves from branches before inserting the branches into holes in logs to "fish" for termites to eat (van Lawick-Goodall, 1971). Certainly such an observation has to be replicated by different scientists in different settings before becoming widely accepted as evidence of toolmaking among chimpanzees, but statistical analysis is not necessary. gt; On the other hand, suppose you wanted to know whether a glass of warm milk at bedtime will help insomniacs get to sleep faster. In this case, the results are not likely to be obvious. You don't expect the warm milk to knock out any of the subjects, or even to help every one of them. The effect of the milk is likely to be small and noticeable only after averaging the time it takes a number of participants to fall asleep (the sleep latency) and comparing that to the average for a (control) group that does not get the milk. Descriptive statistics is required to demonstrate that there is a difference between the two groups, and inferential statistics is needed to show that if the experiment were repeated, it would be likely that the difference would be in the same direction. (If warm milk really has no effect on sleep latency, the next experiment would be just as likely to show that warm milk slightly increases sleep latency as to show that it slightly decreases it.) gt; gt; gt;Variables and Constantsgt; gt; A key concept in the above example is that the time it takes to fall asleep varies from one insomniac to another and also varies after a person drinks warm milk. Because sleep latency varies, it is called a gt;variablegt;. If sleep latency were the same for everyone, it would be a gt;constantgt;, and you really wouldn't need statistics to evaluate your research. It would be obvious after testing a few participants whether the milk was having an effect. But, because sleep latency varies from person to person and from night to night, it would not be obvious whether a particular case of shortened sleep latency was due to warm milk or just to the usual variability. Rather than focusing on any one instance of sleep latency, you would probably use statistics to compare a whole set of sleep latencies of people who drank warm milk with another whole set of people who did not. gt; In the field of physics there are many important constants (e.g., the speed of light, the mass of a proton), but most human characteristics vary a great deal from person to person. The number of chambers in the heart is a constant for humans (four), but resting heart rate is a variable. Many human variables (e.g., beauty, charisma) are easy to observe but hard to measure precisely or reliably. Because the types of statistical procedures that can be used to analyze the data from a research study depend in part on the way the variables involved were measured, we turn to this topic next. gt; gt; gt;Scales of Measurementgt; gt; Measurement is a system for assigning values to observations in a consistent and reproducible way. When most people think of measurement, they think first of physical measurement, in which numbers and measurement units (e.g., minutes and seconds for sleep latency) are used in a precise way. However, in a broad sense, measurement need not involve numbers at all. gt; gt; gt;Nominal Scalesgt; gt; Facial expressions can be classified by the emotions they express (e.g., anger, happiness, surprise). The different emotions can be considered values on a gt;nominal scalegt;; the term gt;nominalgt; refers to the fact that the values are simply named, rather than assigned numbers. (Some emotions can be identified quite reliably, even across diverse cultures and geographical locations; see Ekman, 1982.) If numbers are assigned to the values of a nominal scale, they are assigned arbitrarily and therefore cannot be used for mathematical operations. For example, the gt;Diagnostic and Statistical Manualgt; of the American Psychiatric Association (the latest version is gt;DSM-IVgt;) assigns a number as well as a name to each psychiatric diagnosis (e.g., the number 300.3 designates obsessive-compulsive disorder). However, it makes no sense to use these numbers mathematically; for instance, you cannot average the numerical diagnoses of all the members in a family to find out the average mental illness of the family. Even the order of the assigned numbers is arbitrary; the higher gt;DSM-IVgt; numbers do not indicate more severe diagnoses. gt; Many variables that are important to psychology (e.g., gender, type of psychotherapy) can be measured only on a nominal scale, so we will be dealing with this level of measurement throughout the text. Nominal scales are often referred to as gt;categorical scalesgt; because the different levels of the scale represent distinct categories; each object measured is assigned to one and only one category. A nominal scale is also referred to as a gt;qualitativegt; level of measurement because each level has a different quality and therefore cannot be compared with other levels with respect to quantity. gt; gt; gt;Ordinal Scalesgt; gt; A quantitative level of measurement is being used when the different values of a scale can be placed in order. For instance, an elementary school teacher may rate the handwriting of each student in a class as excellent, good, fair, or poor. Unlike the categories of a nominal scale, these designations have a meaningful order and therefore constitute an gt;ordinal scalegt;. One can add the percentage of students rated excellent to the percentage of students rated good, for instance, and then make the statement that a certain percentage of the students have handwriting that is "better than fair." gt; Often the levels of an ordinal scale are given numbers, as when a coach rank-orders the gymnasts on a team based on ability. These numbers are not arbitrary like the numbers that may be assigned to the categories of a nominal scale; the gymnast ranked number 2 gt;isgt; better than the gymnast ranked number 4, and gymnast number 3 is somewhere between. However, the rankings cannot be treated as real numbers; that is, it cannot be assumed that the third-ranked gymnast is midway between the second and the fourth. In fact, it could be the case that the number 2 gymnast is much better than either number 3 or 4, and that number 3 is only slightly better than number 4 (as shown in Figure 1.1). Although the average of the numbers 2 and 4 is 3, the average of the abilities of the number 2 and 4 gymnasts is not equivalent to the abilities of gymnast number 3. gt; A typical example of the use of an ordinal scale in psychology is when photographs of human faces are rank-ordered for attractiveness. A less obvious example is the measurement of anxiety by means of a self-rated questionnaire (on which subjects indicate the frequency of various anxiety symptoms in their lives using numbers corresponding to never, sometimes, often, etc.). Higher scores can generally be thought of as indicating greater amounts of anxiety, but it is not likely that the anxiety difference between subjects scoring 20 and 30 is going to be exactly the same as the anxiety difference between subjects scoring 40 and 50. Nonetheless, scores from anxiety questionnaires and similar psychological measures are usually dealt with mathematically by researchers as though they were certain the scores were equally spaced throughout the scale. gt; The topic of measurement is a complex one to which entire textbooks have been devoted, so we will not delve further into measurement controversies here. For our purposes, you should be aware that when dealing with an ordinal scale (when you are sure of the order of the levels but not sure that the levels are equally spaced), you should use statistical procedures that have been devised specifically for use with ordinal data. The descriptive statistics that apply to ordinal data will be discussed in the next two chapters. The use of inferential statistics with ordinal data will not be presented until Chapter 21. (Although it can be argued that inferential ordinal statistics should be used more frequently, such procedures are not used very often in psychological research.) gt; gt; gt;Interval and Ratio Scalesgt; gt; In general, physical measurements have a level of precision that goes beyond the ordinal property previously described. We are confident that the inch marks on a ruler are equally spaced; we know that considerable effort goes into making sure of this. Because we know that the space, or interval, between 2 and 3 inches is the same as that between 3 and 4 inches, we can say that this measurement scale possesses the gt;interval propertygt; (see Figure 1.2a). Such scales are based on gt;unitsgt; of measurement (e.g., the inch); a unit at one part of the scale is always the same size as a unit at any other part of the scale. It is therefore permissible to treat the numbers on this kind of scale as actual numbers and to assume that a measurement of three units is exactly halfway between two and four units. gt; In addition, most physical measurements possess what is called the gt;ratio propertygt;. This means that when your measurement scale tells you that you now have twice as many units of the variable as before, you really gt;dogt; have twice as much of the variable. Measurements of sleep latency in minutes and seconds have this property. When a subject's sleep latency is 20 minutes, it has taken that person twice as long to fall asleep as a subject with a sleep latency of 10 minutes. Measuring the lengths of objects with a ruler also involves the ratio property. Scales that have the ratio property in addition to the interval property are called gt;ratio scalesgt; (see Figure 1.2b). gt; Whereas all ratio scales have the interval property, there are some scales that have the interval property but not the ratio property. These scales are called gt;interval scalesgt;. Such scales are relatively rare in the realm of physical measurement; perhaps the best known examples are the Celsius (also known as centigrade) and Fahrenheit temperature scales. The degrees are equally spaced, according to the interval property, but one cannot say that something that has a temperature of 40 degrees is twice as hot as something that has a temperature of 20 degrees. The reason these two temperature scales lack the ratio property is that the zero point for each is arbitrary. Both scales have different zero points (0°C 32°F), but in neither case does zero indicate a total lack of heat. (Heat comes from the motion of particles within a substance, and as long as there is some motion, there is some heat.) In contrast, the Kelvin scale of temperature is a true ratio scale because its zero point represents gt;absolutegt; zero temperature—a total lack of heat. (Theoretically, the motion of internal particles has stopped completely.) gt; Although interval scales may be rare when dealing with physical measurement, they are not uncommon in psychological research. If we grant that IQ scores have the interval property (which is open to debate), we still would not consider IQ a ratio scale. It doesn't make sense to say that someone who scores a zero on a particular IQ test has no intelligence at all, unless intelligence is defined very narrowly. And does it make sense to say that someone with an IQ of 150 is exactly twice as intelligent as someone who scores 75? gt; gt;(Continues...)gt; gt; gt; gt; gt;gt; gt;gt;gt; Excerpted from gt;Explaining Psychological Statisticsgt; by gt;Barry H. Cohengt; Copyright © 2008 by John Wiley & Sons, Ltd. Excerpted by permission of John Wiley & Sons. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.gt;Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.