1 Week 1 Introduction to Statistics and Variables Reading
Introduction
The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives. One of the goals of this class is to help you learn to pay attention to the data all around you, and the statistics produced from them.
In statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample . The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population
Why would we want to sample? Why not just collect the data from the whole population? Populations tend to be very large, like all US residents, or all automobiles. It is very difficult, and time-consuming, to collect data from so many subjects. Maybe you would like to know the average number of siblings of a Fort Lewis student. While, yes, you might be able to track down every single student, this is really not realistic. Instead, you would select a sample of students who would tell you how many siblings that they have, calculate the average, then generalize that average to ALL Fort Lewis students.
Just what is a statistic? Generally speaking, it is a numerical characteristic of a sample. For example, finding an ‘average’ is a statistic of the data that it is calculated from. This course will cover many types of statistics, not so much how to calculate them by hand, but more about how to use technology to get them, as well as how to contextually interpret them. The counterpart to a statistic is called a parameter. A parameter is a numerical characteristic of the population, that can be estimated using a statistic. The average number of siblings of ALL Fort Lewis students would be a parameter. The average number of siblings of the students who were sampled is a statistic.
Variables
A variable is a characteristic or measurement that you wish to study, and collect data about. Recall from Algebra, a variable must be able to change. If we were interested in the average number of siblings of a Fort Lewis student, the variable we wish to measure (and collect data about) is number of siblings. Each subject/student who participates in the study would tell the researcher how many siblings that they have. Be sure not to confuse a variable with a constant. In this study, all of the subjects must be Fort Lewis students in order to participate. Being a Fort Lewis student would be a constant, rather than a variable.
Independent and Dependent Variables
The independent variable has several possible meanings in statistics. If a variable is being controlled by the researcher, it would be known as an independent variable. We also look to see if the independent variable causes, or affects, changes in the dependent variable. The dependent variable changes in response to the independent variable, The independent variable is also known as the explanatory variable; while the dependent variable is also known as the response variable. These alternate vocabulary will be discussed when we get to Correlation and Regression.
A die hard baseball fan wants to see if there is a difference in the average number of hits between Babe Ruth and Hank Aaron. In this situation, the fan would look at the number of hits for each player, where the number of hits depends upon which player it is. The number of hits would be the dependent variable, while the player (Ruth or Aaron) would be the independent variable.
Situations in statistics do not always have two variables. Some of the data that we wish to analyze will only have a single variable. When that is the case, the variable is designated as the dependent (response) variable. Perhaps someone is interested in whether the height of Fort Lewis students, in inches, differs from the national average height of college students. In this situation, there is only one variable. The height of FLC students, in inches, is the dependent variable; this is the only data being collected. The national average is a statistic being used for comparison, a known value, and would not be a variable.
It is also possible to have three or more variables that you are interested in analyzing. While you will come across a few situations in this class that have three or more variables, we will not be doing actual analysis of these situations in this course.
Quantitative and Qualitative Variables
In statistics, it is critical to be aware of which type of data that you are working with. Data can be quantitative, which means that it consists of meaningful numerical values. Number of hits by a baseball player is quantitative data. Data can also be qualitative, sometimes known as categorical. Qualitative data consists of categories or qualities, or sometimes, numbers that have no meaning. Baseball players would be qualitative. Other qualitative data includes hair color, eye color, favorite type of food. Position that you finish in a race, ie 1st, 2nd, 3rd, etc, is actually qualitative. Even though it seems like you are collecting numerical values, those values, 1st, 2nd, 3rd, are not actually meaningful. They are just the categorical name of finishing position.
Identifying/Defining Variables
In this class, you will frequently be asked to identify your variables for various situations. In general, this means that you will have to state the actual variables, classify as quantitative or qualitative, and if appropriate, designate independent and dependent. Going back to the baseball situation which was looking for a difference in the average number of hits between Babe Ruth and Hank Aaron, completely defining the variables would be:
- Independent variable: player (Ruth or Aaron), qualitative.
- Dependent variable: number of runs, quantitative.
Levels of Measurement
Quantitative and qualitative variable classification can further be broken down into four levels of measurement: Nominal, Ordinal, Interval and Ratio.
A designation of nominal is given to a qualitative variable which is simply a name, category or quality. Which type of running shoe you prefer is a nominal measure.
Ordinal measures are also names and categories, ie qualitative, but they also have a specific order to those names. An example of this would be Olympic medals. Bronze, silver and gold are categories, but there is a clear order to these in that gold is the highest achievement, then silver, then bronze.
Numerical, ie quantitative, variables are classified as either interval or ratio.
The interval designation is the level of measurement when the numerical values have meaningful distance between them. Ratio data goes beyond just meaningful distance between the values, and also includes a true zero to the data AND meaningful magnitude. Temperature measurements in Celsius and Fahrenheit are interval measures, because they are meaningful numerical values, but lack both a true zero and magnitude. 0 degrees Celsius does not mean the absence of temperature, nor is 100 degrees twice as warm as 50 degrees. The volume in a can of sparkling water measured in mL is ratio data. 0 mL is the absence of volume in a can, ie the can is empty. Also, a 16 oz can has twice as much volume as a can with only 8 oz.
One important thing to remember is that when asked a variable’s level of measurement, there is only ONE response. If the variable qualifies to be ratio, then it is only designated as ratio. If a variable is ordinal, it is only designated as ordinal.
| Type of Variable | Level of Measurement | Definition | Examples |
|---|---|---|---|
| Qualitative | Nominal | Assigns name, category, quality | Type of Running Shoe, Color of Eye |
| Qualitative | Ordinal | Assigns names and categories, but adds order to the categories
|
Olympic Medals (Gold, Bronze, Silver), Military Rank, Letter Grades |
| Quantitative | Interval | Numerical values have meaningful distances. Zero is meaningless | Temperature in Fahrenheit and Celsius |
| Quantitative | Ratio | Zero has meaning, magnitude is meaningful
|
Volume in ounces, Height in inches, Weight in pounds |
Quantitative Variables: Discrete or Continuous?
Finally, quantitative variables may also be characterized as discrete or continuous. These two designations are an either/or, never both, when classifying a numerical variable. Discrete data can only take on certain values, and includes things that can be counted. Continuous data can be measured, and always includes some sort of units. The number of students in your Statistics class is discrete data. You would count the students; there would never be part of a student. How many plants you have ever owned is also discrete. The height of a person, in inches, is continuous data – you would measure this variable.
For those with a background in Algebra, discrete vs continuous can become confusing. In Statistics, continuous does not mean quite the same thing as in Algebra. Regarding the height of a person being continuous, this is not to say that height of a person can get infinitely large or small, but rather can be measured down to the part of a part of a part of an inch. Also, do not be fooled by measures that people regularly round to whole numbers, like their age, weight or height. One might actually be 65.136937 inches, but simply say that they are 65 inches tall. Just because data is reported as whole numbers does not automatically mean that the variable is discrete.
Lurking Variables
Two variables may be related, but this does not guarantee that one variable is influencing the other. In a study of weight gain in cats, researchers found that there was a strong connection between activity level and weight gain, indicating that less activity corresponds to greater weight gain. But, there are other factors that might be at play here. Age of the cat, or species may be lurking variables. A lurking variable is not an independent/explanatory variable that we are studying, BUT it is a variable that is having a confounding effect on the variables that we are studying. We would need to design our study such that age is consistent among the cats, and sample from cats of the same species in order to eliminate age and species as lurking variables. Otherwise, age and species may be having some effect on cat weight gain, when we are really just interested in a connection between activity level and weight gain. This Lurking Variables YouTube video [5:18] gives a few more examples.
Student Course Learning Objectives
- Define basic statistics vocabulary (e.g., levels of measurement (nominal, ordinal, interval, ratio), discrete vs. continuous variables, descriptive vs. inferential statistics, sample vs. population, independent vs. dependent variable, explanatory vs. response variable, confounding variables, experimental vs. observational)
Attributions
Adapted from “Week 1 Introduction to Statistics and Variable Reading” by Sherri Spriggs and Sandi Dang is licensed under CC BY-NC-SA 4.0
