Statistics

The Essentials for Research

Henry E. Klugh

• Publisher: Psychology Press
• Year: 2013
• ISBN:
  9781134928699
• Edition: 3
• Topic: Psicología
• Subtopic: Historia y teoría en psicología

Published in 1986, Statistics is a valubale contribution to the field of Research Methods/Stats.

1

Introduction

1.1 Advice to the Student

Many students in the behavioral sciences approach their first course in statistics wishing they had majored in some other discipline—any other discipline not requiring them to study statistics. They are convinced that their “aptitude for math” is low, and they are prepared to find statistics difficult. Statistics is difficult for many students but not, as a rule, because they have a low aptitude for mathematics. Students who have little or no college work in mathematics suffer from a serious misconception about the rate at which they should be able to read mathematical material. Accustomed to reading a 50-page assignment for other courses in two or three hours, they discover that they have spent that much time before they understand six pages of statistics. When this happens, they become absolutely convinced that they have no aptitude for mathematics and, in despair, they drop statistics and change their majors! If you have had similar thoughts, the following paragraphs are especially for you.

Not even mathematicians read mathematics as rapidly as they read other material. Of course, this depends on the material and on the individual, but even professional mathematicians read unfamiliar mathematics at a considerably slower pace than they read anything else. When you require a great deal of time to understand a page of mathematics, it does not reflect unfavorably on your intelligence or mathematical aptitude. You should expect to read a statistics textbook much more slowly than you read other textbooks, and you should expect to reread some sections a number of times before the relationships discussed become clear.

One reason for slower reading in a statistics course is that complex ideas are communicated by the use of unfamiliar symbols. In most other courses, you already know the meaning of the words by which any new ideas are communicated. Your problem is to understand and then to remember a novel thought communicated by a new arrangement of largely familiar symbols. In statistics, however, many of the symbols and most of the concepts are entirely new! You must begin by learning this new and fairly complex vocabulary of symbols before you can understand the concepts communicated by that vocabulary. For this reason you should make sure you know the meaning of each new symbol or term before you read beyond the section in which it is introduced. And you should expect the study of statistics to take more time, page for page, than you must devote to your other courses. It is very important that you see this process as a challenge which you can meet. The material is much like a crossword puzzle, a chess problem, or a challenging bridge hand. Certainly a portion of your task is to remember, but in this course, that is far less important than to understand, to comprehend. In many college courses understanding and comprehension are automatic. The tough task is remembering specific facts. In statistics the tables are turned; understanding is the primary task. When that is accomplished, retention will be almost automatic!

You can check your comprehension by answering the questions at the end of the chapters, and by reviewing the adequacy of your answers in the answer section before going on to the next question. Answers for most of the problems are supplied, and the procedures by which certain answers are obtained have also been included in the answer section. If you cannot answer one of the questions you should go back and reread the appropriate section of that chapter. Above all you must study the material regularly, but preferably not for more than a few hours at a time. Finding yourself a chapter behind on the day before the test is not a position from which you can recover by an all-night study session. If you are willing to exert consistent effort, you will probably finish the course with much more respect for your “mathematical aptitude” than you had when you began.

1.2 On Calculators

One piece of equipment that will make your statistics course much easier is the pocket calculator, which sells for as little as $15.00, and is really indispensable for much of the homework you will be assigned. There are, of course, many excellent models and more are appearing all the time so we will not make any recommendations here. Your instructor will have some advice about calculators.

Of course if you have access to a small computer with an appropriate statistical software package, life will be even more pleasant. If this isn’t available, don’t worry. All calculations can be completed within a reasonable time on a good electronic calculator.

1.3 Why Statistics?

All sciences, including psychology, try to describe and ultimately understand relationships between the empirical events (observations) in their disciplines. In some areas of science (notably physics and chemistry), but also in some sections of psychology, the relationships between these events may be clear cut and easy to demonstrate. For example, the length of time it will take a 1-cubic-inch marble to fall 4 feet can be determined with a fairly high degree of accuracy. If air density is kept constant, and our instruments are in order, we can probably obtain almost exactly the same result with all marbles of similar dimensions. In this example from physics one must consider the density of the medium and the shape of the marble, but for all practical purposes that ends the list of variables that might affect the outcome.

On the other hand, we might wish to know the speed with which a rat will traverse a 4-foot alley for food reward. We can set up instruments for measuring elapsed time which are just as sophisticated as those used in the physics experiment, but it is quite unlikely that the psychologist’s rats will produce the consistent speeds produced by the physicist’s marbles. The behavioral scientist has a great many more variables to control. Of course, the rats to be compared should all be equally deprived of food, all of the same sex, age, and weight, all receive the same amount and type of food reward on earlier trials, and all be housed under identical conditions. If we carefully observe all of these controls, and then compare the running times of two rats chosen by lot, we shall almost certainly find the times to be different; not quite as different as they would have been without the controls, but different nevertheless.

In Table 1.1, in the theoretical column, we have recorded the running times one might expect for a group of “identical” rats, if these were obtainable and, in the observed column, the running time of real rats as they might be recorded in a real experiment.

Table 1.1
Time to Traverse an Alley Maze

Rat	Theoretical Rats	Observed Rats
1	4 sec.	6 sec
2	4 ”	8 ”
3	4 ”	4 ”
4	4 ”	3 ”
5	4 ”	2 ”

Even if we have exerted every effort to hold constant the unwanted influences on running time, it is still quite safe to assume that we have not controlled them all. Some rats may have been handled a bit more roughly than others; some may have had a fight with their cage mate just before running the alley; one may have noticed an attractive (or repulsive) odor left by the previous occupant of the start box and adjusted his time accordingly. In short, the study of behavior often involves a host of variables, not all of which can be controlled, that act to disguise the relationships between the variables under investigation.

All scientific observations, even those of physicists, contain a true component and an error component. The true component is equivalent to the theoretical running times of Table 1.1, and the error component is the sum of all the chance, or randomly operating uncontrolled variables that, when added to the true component, give rise to each entry in the observed column. This error component often tends to disguise relationships between events just as static tends to disguise intelligible sound from a radio.

There are two ways to reduce the effect of error: experimentally, by careful laboratory procedures; and statistically, by increasing the number of observations and manipulating the data so that the relationships will be apparent in spite of the random or chance error. Statistics, as a discipline, is concerned with this latter process. In its applied form, as we shall study it in this text, it is concerned with describing and drawing inferences from many observations; observations that are ordinarily translated into measurements or counts.

The study of statistics may be divided into two broad areas. One of these areas is called inferential statistics because it deals with inferences about the true nature of the relationships between variables in spite of the ever present chance or error component in their measurement. Most of this book is devoted to inferential statistics. Before we can infer anything from observations, however, they must be described in a systematic fashion. This branch, called descriptive statistics, shows us efficient ways to describe and summarize data, and consequently how to present it in the most usable form. It is this aspect of statistics that we now discuss.

2

Graphing Distributions

In this chapter we discuss the graphical presentation of data, but first we comment briefly on the nature of the data with which the scientist works.

2.1 Operational Definitions

An experiment, in its simplest form, is designed to investigate the effect of one variable upon another. A variable may be defined as any property on which events or objects can take different values. For example, although they represent rather different kinds of variables, IQ, height, sex, and family size are all variables. Scientific convention uses the term “independent variable” to designate any variable presumed to exert the effect, and the term “dependent variable” to designate the variable presumably affected. If we investigate the effect of hunger on activity, hunger is the independent variable and activity is the dependent variable.

If you keep your subjects away from food and observe any systematic changes in their tendency to be active, you have some of the elements of an experiment. You might take notes on the behavior of your subjects, and then summarize your observations in a written description of their behavior. Unfortunately, another investigator conducting the same experiment might write a different report, not necessarily because of differences in the behavior of the animals but, perhaps, because of differences between you and the other investigator regarding the kinds of behavior each considered to be indicative of “activity.”

We can increase the objectivity and hence the reliability of such an experiment if we define hunger and activity in a way that permits their measurement. If we define hunger by specifying the operations used to produce or measure it, the definition is called an operational definition. We can define the degree of hunger operationally in terms of the number of hours since food was last available to the animal. Thus, by definition, an animal deprived of food for 24 hours is “hungrier” than one which has been deprived of food for 6 hours. Notice that such a definition does not describe the internal stimuli produced by the absence of food, nor does it describe the sensations presumably endured by a hungry animal. In fact, there are a number of ways in which such a definition is deficient, but it is an operational definition; it specifies the operations by which “hunger” is produced. If we accept this operational definition of the independent variable we can form different “hunger” subgroups by depriving some animals of food for 6 hours, some for 12 hours, and some for 24 hours.

Similarly, activity can be operationally defined as the number of rotations of an activity wheel made by the animal during a 5-minute test period. Each subject in the different “hunger” subgroups can then be given an activity score, and average activity scores can be compared among the hunger subgroups. If different experime...