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The Jonathan Dolhenty Archive |
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PAGE ONE by Jonathan Dolhenty, Ph.D. Index:
At least an elementary knowledge of statistics is necessary for every critical thinker in today's world. Statistical information of all sorts is presented almost daily in the media. Statistical method is a technique used to obtain, analyze and present numerical data. The elements of statistical technique include the following:
The characteristics and limitations of statistical methods should be kept in mind. Statistical method is the only means for handling large masses of numerical data. Statistical technique applies only to data which are reducible to quantitative form, that is, the data must be capable of being expressed as measurements of some sort. Statistical technique is objective; the results, however, cannot but be affected by the necessarily subjective interpretation. Statistical technique is the same for the social sciences as for the physical sciences. Statistical technique is commonly used in economics, education, sociology, psychology, biology, chemistry, astronomy, and so on. Method and technique apply alike to these divergent fields. A basic understanding of statistics demands a basic understanding of what are called variables and what are called constants. A statistical study is based on some observation of behavior wherein the objects of study change from time to time and, in the case of human subjects, from person to person. Traits which are capable of variation are called variables. Examples of such traits include intelligence, height, actions, opinions, political party membership, church attendance, hair color, and so forth. Statistical studies are based on the relationships between variables. Population is a statistical term that refers to the entire group of observations under study. (Sometimes the word universe is used.) Any study of a population will describe that population in terms of characteristics that the members have in common as well as those that vary. Those characteristics which do not vary from individual to individual within the population being studied are called constants. A distinction may be made between continuous variables and discrete variables. A continuous variable may take any value within a defined range of values. Between any two values of the variable an indefinitely large number of in-between values may occur. Examples of continuous variables include intelligence, height, weight, and chronological time. A discrete variable can take specific values only. For instance, size of family is a discrete variable. A family may include 2, 3, or 4 children, but values between these are not possible. A family cannot have 2 and 1/2 children! Other examples of discrete variables are the population of a city and the score of a baseball game. Have you ever heard of a baseball game with the score of 5 1/2 to 7 1/4? Some statistical procedures are appropriate for use with continuous variables, while others are appropriate only for use with discrete variables. A distinction may also be made between variables which vary in quality and variables which vary in quantity. Eye color and degree of aggressiveness, for example, are qualitative variables, while intelligence and temperature are quantitative variables.
Measurement is the assignment of numbers to objects or events according to certain prescribed rules. It is common practice to use four kinds of scales to describe the varying levels of measurement. Each type of scale has a different set of rules. Nominal Scales This is the simplest form of measurement; an object is simply placed into a category according to some means of classification. The object or event either is or is not a member of the category being considered. For example, individuals may be classified according to the color of their eyes. Each individual is either in or out of a specific category of eye color. Dogs may be classified according to the categories of hunting, working, herding, and so forth. Each dog is in a specific category and out of any other category. A nominal variable is a characteristic of the members of a group defined by an operation which permits the making of statements only of equality or difference. We may state that one member is the same or different from another member with respect to the characteristic under consideration. A specific dog, for instance, is either the same or different from some other dog as to the category of hunting dogs. Ordinal Scales Ordinal scales permit the establishment of orders among categories. While nominal scales show that things are different, ordinal scales show the direction of the difference. Statements about ordinal variables include not only statements about "same as" and "different from," but also statements of the kind "greater than" or "less than." Thus, individuals can be ranked, for instance, according to the characteristic of weight. We cannot, however, make a statement regarding the amount of difference between the rankings. Four people may be ranked according to their cooperativeness, but the individual who ranks second is not twice as cooperative as individual number 1. Again, we can rank individuals according to height, but individual number 2 is not twice as tall as individual number 1! Nominal and ordinal scales have important uses, particularly in studies about human behavior. The statistical techniques which have been developed for use with these scales are called nonparametric statistics. Interval Scales Interval scales have equal intervals between the units of measure. For example, the temperature scale on a thermometer is an interval scale. A temperature of 60 degrees F. is halfway between 50 degrees F. and 70 degrees F. It should be noted that interval scales do not have a true "zero" point. A zero point, however, may be arbitrarily defined as a convenience, such as we do with thermometers and calendars and intelligence scores (it is meaningless to think about an individual with zero intelligence). A word of warning about interval scales should be given here. Let's consider three temperatures: "A" = 12 degrees, "B" = 24 degrees, and "C" = 36 degrees. It is proper to that the difference between temperatures "A" and "B" is equal to the difference between temperatures "B" and "C." It is also proper to say that the difference between "A" and "C" is twice the difference between "A" and "B" or "B" and "C." But it is NOT proper to say that "B" has twice the temperature of "A" or that "C" has three times the temperature of "A." If the temperature outside today was 70 degrees, and the temperature yesterday was 35 degrees, we would not ordinarily say that today it is twice as hot as yesterday. An interval variable, then, is a characteristic defined by an operation which permits the making of statements of equality of intervals, in addition to statements of sameness or difference or greater than or less than. Other examples of interval variables are calendar time and the scores on intelligence tests. Ratio Scales Ratio scales have the same qualities as interval scales, plus they have the property of an absolute zero point. Therefore, a ratio variable is a property defined by an operation which permits the making of statements of equality of ratios in addition to all other kinds of statements already discussed. This means that one variate measurement may be said to be double or triple another, and so on. An absolute zero is always implied. Variables such as height, weight, and age are ratio variables and can be expressed on ratio scales. A person of 50 years of age is twice the age of a person of 25. A person weighing 300 pounds is three times the weight of a person weighing only 100 pounds. A word of warning, however. This cannot be said of IQ scores! We cannot say that a person with an IQ of 100 is twice as intelligent as a person with a score of 50. "Zero intelligence" cannot be defined! The essential difference between a ratio and an interval variable is that for the ratio variable the measurements are made from a true zero point, whereas for the interval variable the measurements are made from an arbitrarily defined zero point. Statistical procedures used with ratio scales will usually be the same as those used with interval scales. These techniques are referred to as parametric statistics.
Once the data from a study or survey are collected, they must be organized. This unorganized information is sometimes referred to as the raw data. Only when the data are organized, can they then be analyzed. We will consider only three ways of classifying data: frequency distributions, simple ranking, and percentile ranking. These are the ones most ordinary people are somewhat acquainted with and these are commonly used in presenting the results of school test scores and opinion surveys. Frequency Distribution A frequency distribution arranges a collection of measures in graphic form to indicate the frequency of occurrence of each value. The number of times a particular score value occurs is a frequency.
Simple Ranking Using a frequency distribution, additional analysis may be performed through ranking the data. Each score is given a position from high to low, with 1 indicating the highest rank. Sometimes there will be more than one incident of the same score, that is, two or more scores may occur in one position. In such a case, the scores are averaged to determine their rank.
Percentile Ranking Parents often see their child's standardized test score given as a percentile rank. It is important to understand what this means. A score on a test, or any other measuring instrument for that matter, has little meaning unless it is related to other scores. The percentile rank of a raw score is the percentage of scores below the specific score being considered. The percentile rank is a relative rank, a rank order score based on a scale of 100. Thus, the percentile rank is a point on a scale ranging from 1 to 100. An example may help. If 75 percent of a group of students score below a certain student, that student's percentile rank is 75. The centile is the point on a raw score scale which corresponds to a given percentile rank. An IQ score of 100, for example, is normally the fiftieth centile. The fiftieth centile is 100. The percentile rank of an IQ score of 100 is normally 50.
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