Introduction to Measurement and Statistics

"Statistics can be fun or at least they don't need to be feared." Many folks have trouble believing this premise. Often, individuals walk into their first statistics class experiencing emotions ranging from slight anxiety to borderline panic. It is important to remember, however, that the basic mathematical concepts that are required to understand introductory statistics are not prohibitive for any university student. The key to doing well in any statistics course can be summarized by two words, "KEEP UP!". If you do not understand a concept--reread the material, do the practice questions, and do not be afraid to ask your professor for clarification or help. This is important because the material discussed four weeks from today will be based on material discussed today. If you keep on top of the material and relax a little bit, you might even find you enjoy this introduction to basic measurements and statistics.

With that preface out of the way, we can now get down to the business of discussing, "What do the terms measurement and statistic mean?" and "Why should we study measurement and statistics?"

What is a Statistic?

Statistics are part of our everyday life. Science fiction author H. G. Wells in 1903 stated, ""Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write." Wells was quite prophetic as the ability to think and reason about statistical information is not a luxury in today's information and technological age. Anyone who lacks fundamental statistical literacy, reasoning, and thinking skills may find they are unprepared to meet the needs of future employers or to navigate information presented in the news and media On a most basic level, all one needs to do open a newspaper, turn on the TV, examine the baseball box scores, or even just read a bank statement (hopefully, not in the newspaper) to see statistics in use on a daily basis.

Statistics in and of themselves are not anxiety producing. For example, most individuals (particularly those familiar with baseball) will not experience anxiety when a player's batting average is displayed on the television screen. The "batting average" is a statistic but as we know what it means and how to interpret it, we do not find it particularly frightening. The idea of statistics is often anxiety provoking simply because it is a tool with which we are unfamiliar. Therefore, let us examine what is meant by the term statistic; Kuzma (1984) provides a formal definition: :

A body of techniques and procedures dealing with the collection, organization, analysis, interpretation, and presentation of information that can be stated numerically.

Perhaps an example will clarify this definition. Say, for example, we wanted to know the level of job satisfaction nurses experience working on various units within a particular hospital (e.g., psychiatric, cardiac care, obstetrics, etc.). The first thing we would need to do is collect some data. We might have all the nurses on a particular day complete a job satisfaction questionnaire. We could ask such questions as "On a scale of 1 (not satisfied) to 10 (highly satisfied), how satisfied are you with your job?". We might examine employee turnover rates for each unit during the past year. We also could examine absentee records for a two month period of time as decreased job satisfaction is correlated with higher absenteeism. Once we have collected the data, we would then organize it. In this case, we would organize it by nursing unit.

Absenteeism Data by Unit in Days
Psychiatric Cardiac CareObstetrics
69 4
Mean = 594

Thus far, we have collected our data and we have organized it by hospital unit. You will also notice from the table above that we have performed a simple analysis. We found the mean (you probably know it by the name "average") absenteeism rate for each unit. In other words, we added up the responses and divided by the number of them. Next, we would interpret our data. In this case, we might conclude that the nurses on the cardiac care unit are less satisfied with their job as indicated by the high absenteeism rate (We would compare this result to the analyses of our other data measures before we reached a conclusion). We could then present our results at a conference, in a journal, or to the hospital administration. This might lead to further research concerning job satisfaction on cardiac care units or higher pay for nurses working in this area of specialization.

This example clarifies the process underlying statistical analyses and interpretation. The various techniques and procedures used in the process described above are what make up the content of this chapter. Thus, we will first learn a little bit about the process of data collection/research design. Second, we will examine the use and interpretation of basic statistical analyses used within the context of varying data and design types. And finally, we will examine the process of data presentation.

To further our understanding of the term statistics, it is important to be aware that statistics can be divided into two general categories: descriptive and inferential statistics. Each of these will be discussed below.

Descriptive statistics are used to organize or summarize a particular set of measurements. In other words, a descriptive statistic will describe that set of measurements. For example, in our study above, the mean described the absenteeism rates of five nurses on each unit. The U.S. census represents another example of descriptive statistics. In this case, the information that is gathered concerning gender, race, income, etc. is compiled to describe the population of the United States at a given point in time. A baseball player's batting average is another example of a descriptive statistic. It describes the baseball player's past ability to hit a baseball at any point in time. What these three examples have in common is that they organize, summarize, and describe a set of measurements.

Inferential statistics use data gathered from a sample to make inferences about the larger population from which the sample was drawn. For example, we could take the information gained from our nursing satisfaction study and make inferences to all hospital nurses. We might infer that cardiac care nurses as a group are less satisfied with their jobs as indicated by absenteeism rates. Opinion polls and television ratings systems represent other uses of inferential statistics. For example, a limited number of people are polled during an election and then this information is used to describe voters as a whole.

What is Measurement ?

Normally, when one hears the term measurement, they may think in terms of measuring the length of something (e.g., the length of a piece of wood) or measuring a quantity of something (ie. a cup of flour).This represents a limited use of the term measurement. In statistics, the term measurement is used more broadly and is more appropriately termed scales of measurement. Scales of measurement refer to ways in which variables/numbers are defined and categorized. Each scale of measurement has certain properties which in turn determines the appropriateness for use of certain statistical analyses. The four scales of measurement are nominal, ordinal, interval, and ratio.

Nominal: Categorical data and numbers that are simply used as identifiers or names represent a nominal scale of measurement. Numbers on the back of a baseball jersey and your social security number are examples of nominal data. If I conduct a study and I'm including gender as a variable, I may code Female as 1 and Male as 2 or visa versa when I enter my data into the computer. Thus, I am using the numbers 1 and 2 to represent categories of data.

Ordinal: An ordinal scale of measurement represents an ordered series of relationships or rank order. Individuals competing in a contest may be fortunate to achieve first, second, or third place. first, second, and third place represent ordinal data. If Roscoe takes first and Wilbur takes second, we do not know if the competition was close; we only know that Roscoe outperformed Wilbur. Likert-type scales (such as "On a scale of 1 to 10, with one being no pain and ten being high pain, how much pain are you in today?") also represent ordinal data. Fundamentally, these scales do not represent a measurable quantity. An individual may respond 8 to this question and be in less pain than someone else who responded 5. A person may not be in exactly half as much pain if they responded 4 than if they responded 8. All we know from this data is that an individual who responds 6 is in less pain than if they responded 8 and in more pain than if they responded 4. Therefore, Likert-type scales only represent a rank ordering.

Interval: A scale that represents quantity and has equal units but for which zero represents simply an additional point of measurement is an interval scale. The Fahrenheit scale is a clear example of the interval scale of measurement. Thus, 60 degree Fahrenheit or -10 degrees Fahrenheit represent interval data. Measurement of Sea Level is another example of an interval scale. With each of these scales there are direct, measurable quantities with equality of units. In addition, zero does not represent the absolute lowest value. Rather, it is point on the scale with numbers both above and below it (for example, -10degrees Fahrenheit).

Ratio: The ratio scale of measurement is similar to the interval scale in that it also represents quantity and has equality of units. However, this scale also has an absolute zero (no numbers exist below zero). Very often, physical measures will represent ratio data (for example, height and weight). If one is measuring the length of a piece of wood in centimeters, there is quantity, equal units, and that measure cannot go below zero centimeters. A negative length is not possible.

The table below will help clarify the fundamental differences between the four scales of measurement:

Indications Difference Indicates Direction of Difference Indicates Amount of Difference Absolute Zero
Ordinal XX
Interval XXX

You will notice in the above table that only the ratio scale meets the criteria for all four properties of scales of measurement.

Interval and Ratio data are sometimes referred to as parametric and Nominal and Ordinal data are referred to as nonparametric. Parametric means that it meets certain requirements with respect to parameters of the population (for example, the data will be normal--the distribution parallels the normal or bell curve). In addition, it means that numbers can be added, subtracted, multiplied, and divided. Parametric data are analyzed using statistical techniques identified as Parametric Statistics. As a rule, there are more statistical technique options for the analysis of parametric data and parametric statistics are considered more powerful than nonparametric statistics. Nonparametric data are lacking those same parameters and cannot be added, subtracted, multiplied, and divided. For example, it does not make sense to add Social Security numbers to get a third person. Nonparametric data are analyzed by using Nonparametric Statistics.

As a rule, ordinal data is considered nonparametric and cannot be added, etc.. Again, it does not make sense to add together first and second place in a race--one does not get third place. However, many assessment devices and tests (e.g., intelligence scales) as well as Likert-type scales represent ordinal data but are often treated as if they are interval data. For example, the "average" amount of pain that a person reports on a Likert-type scale over the course of a day would be computed by adding the reported pain levels taken over the course of the day and dividing by the number of times the question was answered. Theoretically, as this represents ordinal data, this computation should not be done.

As stated above, many measures (e.g,. personality, intelligence, psychosocial, etc.) within the psychology and the health sciences represent ordinal data. IQ scores may be computed for a group of individuals. They will represent differences between individuals and the direction of those differences but they lack the property of indicating the amount of the differences. Psychologists have no way of truly measuring and quantifying intelligence. An individual with an IQ of 70 does not have exactly half of the intelligence of an individual with an IQ of 140. Indeed, even if two individuals both score a 120 on an IQ test, they may not really have identical levels of intelligence across all abilities. Therefore, IQ scales should theoretically be treated as ordinal data.

In both of the above illustrations, the statement is made that they should be theoretically treated as ordinal data. In practice, however, they are usually treated as if they represent parametric (interval or ratio) data. This opens up the possibility for use of parametric statistical techniques with these data and the benefits associated with the use of techniques.

Why Study Statistics?

Hopefully, the discussion above has helped you to understand a little better what the terms measurement and statistics mean. However, you may still be wondering "Why do I need to learn statistics?" or "What future benefit can I get from a statistics class?" Well, since you asked--there are five primary reasons to study statistics:

The first reason is to be able to effectively conduct research. Without the use of statistics it would be very difficult to make decisions based on data collected from a research project. For example, in the study cited above, is the difference in recorded absenteeism between psychiatric and obstetrics nurses large enough to conclude that there is meaningful difference in absenteeism between the two units? There are two possibilities: The first possibility is that the difference between the two groups is a result of chance factors. In reality, the two jobs have approximately the same amount of absenteeism. The second possibility is that there is a real difference between the two units with the psychiatric unit demonstrating that these nurses miss more work. Without statistics we have no way of making an educated decision between the two possibilities. Statistics, however, provides us with a tool with which to make an educated decision. We will be able to decide which of the two possibilities is more likely to be true as we base our decision on our knowledge of probability and inferential statistics.

A second point about research should be made. It is extremely important for a researcher to know what statistics they want to use before they collect their data. Otherwise data might be collected that is not interpretable. Unfortunately, when this happens it results in a loss of data, time, and money.

Now many a student may by saying to themselves: "But I never plan on doing any research." Although you may never plan to be involved in research, research may find its way into your life. Certainly, if you decide to continue your education and work on a masters or doctoral degree, involvement in research will result from that decision. Secondly, more and more work places are conducting internal research or are part of broader research studies. Thus, you may find yourself assigned to one of these studies. finally, many classes on the undergraduate level will require you to conduct research (for example, a research methods course). In each of these instances, knowledge of measurements and statistics will be invaluable.

The second reason to study statistics is to be able to read journals. Most technical journals you will read contain some form of statistics. Usually, you will these statistics in something called the results section. Without an understanding of statistics, the information contained in this section will be meaningless. An understanding of basic statistics will provide you with the fundamental skills necessary to read and evaluate most results sections. The ability to extract meaning from journal articles and the ability to critically evaluate research from a statistical perspective are fundamental skills that will enhance your knowledge and understanding in related coursework.

The third reason is to further develop critical and analytic thinking skills. Most students completing high school and introductory undergraduate coursework have at their disposal a variety of critical thinking and analytic skills. The study of statistics will serve to enhance and further develop these skills. To do well in statistics one must develop and use formal logical thinking abilities that are both high level and creative.

The fourth reason to study statistics is to be an informed consumer. Like any other tool, statistics can be used or misused. Yes, it is true that some individuals do actively lie and mislead with statistics. More often, however, well meaning individuals unintentionally report erroneous statistical conclusions. If you know some of the basic statistical concepts, you will be in a better position to evaluate the information you have been given.

The fifth reason to have a working knowledge of statistics is to know when you need to hire a statistician. Most of us know enough about our cars to know when to take it into the shop. Usually, we don't attempt the repair the car ourselves because we do not want to cause any irreparable damage. Also, we try to know enough to be able to carry on an intelligible conversation with the mechanic (or we take someone with us who can) to insure that we do not get a whole new engine (big bucks) when all we need is a new fuel filter (a few bucks). We should be the same way about hiring a statistician. Conducting research is time consuming and expensive. If you are in over your statistical head, it does not make sense to risk an entire project by attempting to compute the data analyses yourself. It is very easy to compute incomplete or inappropriate statistical analysis of one's data. As with the mechanic discussed above, it is also important to have enough statistical savvy to be able to discuss your project and the data analyses you want computed with the statistician you hire. In other words, you want to be able to make sure that your statistician is on the right track.

To summarize, the five reasons to study statistics are to be able to effectively conduct research, to be able to read and evaluate journal articles, to further develop critical thinking and analytic skills, to act as an informed consumer, and to know when you need to hire outside statistical help.

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