Chapter 1 - Welcome to Statistics, A Modeling Approach

1.1 Welcome to Statistics!

Welcome to Statistics and Data Science: A Modeling Approach. Statistics is the study of variation. It is the tools and concepts that have been developed, over centuries, to help us understand variation. There is a lot of variation in the world. As statisticians, we find ways to turn variation in the world into variation in data, and then analyze that data to deepen our understanding of the world. (See figure, from Wild, C. (2006). The concept of distribution. Statistics Education Research Journal, 5(2), 10-26.)

Learning statistics can feel hard. It takes sustained effort over long periods of time, whether you are a beginner or a professional statistician! Even those with substantial statistics experience are always learning new things, and deepening their understanding. In this course, we want to get you started along the pathway to understanding.

The good news is: even though learning statistics feels hard at times, anyone can do it - including you! When it feels hard, that just means you are working on challenging concepts, not that you aren’t capable of mastering them. Don’t forget this.

At the end of the course you will understand more than you do now, and hopefully that will be useful to you. As you will see throughout the text, understanding the basics of statistics and data analysis can help you understand the world around you, and also be relevant to many jobs that you may hold in the future. Learning about statistics can empower you to achieve your goals and pursue your interests.

Building Flexible Knowledge

Most people have learned something about statistics before they take a course in it. Many of you have even taken whole courses in statistics before this one. If you have, you have probably heard about some or all of these things: mean, variance, standard deviation, t test, p, F, ANOVA, regression, chi-square, normal distribution, z-score, and so on.

With such a long list, it’s no surprise that many students see remembering as the most challenging part of learning statistics. But actually, remembering is not the most challenging part. Understanding is the most challenging part. Even if you remember what all these things are, if you don’t understand how it all fits together you will probably forget it all as soon as you are done with the final exam. We don’t want that to happen!

We will discuss lots of these things you have heard of or studied before. But instead of emphasizing their particularity—how each is different from the other—we will work on understanding their coherence—how they are all connected together into a system of thinking. In other words, our goal is to help you see the big picture rather than the individual details. The details will come later, and more easily, if you have an overall framework for understanding where everything fits.

Experts in a domain are distinguished from novices not by the number of things they know, but by the robust connections that tie these things together (Fries et al, 2020). These connections are what make knowledge flexible and transferable; they are what prepare you for solving new problems or taking new courses in the field.

Our goal is to help you understand the big ideas that you can use to understand and interconnect the other ideas and procedures that we will study. This will take time, but it’s worth it!

The Statistical Model

The big idea that unites all of this together is the concept of statistical model. We don’t assume that you know what a statistical model is. Instead, we expect your understanding of this powerful idea to increase gradually throughout the course. Statistical models help us in three main ways.

First, they help us to understand patterns in data and where they come from, or, what we will call the Data Generating Process (or DGP for short). The DGP is the process that causes variation, which we will discuss a lot more later in the course.

Second, they help us to predict what will happen in the future. Of course we can’t really predict the future very well—we aren’t psychic, and you probably aren’t either. But, using statistical models, we can make better predictions than we could without them, even if they can be improved. Sometimes this is very useful. When Netflix recommends a movie you might like, they use a statistical model. They may be wrong sometimes, but they do better than just random guessing!

Finally, statistical models can help you improve the functioning of complex systems. In situations where everything seems to vary, and where the variation seems overwhelming, you can still use statistical models to help you identify changes you can make in one variable that will improve some outcome you are interested in. Some hospitals, for example, use statistical models to help reduce the time patients spend waiting to see a doctor.

Simple + Profound

Jordan Ellenberg in his book How Not to Be Wrong (Ellenberg, J. (2015). How not to be wrong: The power of mathematical thinking. Penguin) provides a nice description of where the content of his book will focus, which applies equally well to this course.

Math, he writes, can be organized along two dimensions (see figure from Ellenberg, above). First, there is simple math and complicated math. \(1+2=3\) is simple math, whereas this is more complicated:

\[f(x) = \int h(x)\, dx=g(x)\]

The other dimension is from shallow to profound. Although the previous two examples differ in terms of how complicated they are, both are relatively shallow in their import.

Professional mathematicians spend their time working on ideas that are complicated and profound (upper right quadrant of the table). But mathematical ideas that are profound do not have to be complicated. Some ideas are simple, yet profound.

For example, the Pythagorean theorem, \(a^2 + b^2 = c^2\), is often taught in geometry courses to help students solve for one of the sides of a triangle. But there is a profoundness to the Pythagorean theorem that has nothing to do with solving for anything. If you take a right triangle, ANY right triangle, the area of squares created from the two smaller sides will equal the area of the square created from the hypotenuse.

Statistics, especially the statistics we will study in this course, falls in this quadrant—simple and profound. The ideas are actually pretty simple, especially when you look past the surface features to the big organizing concepts behind them. But they are powerful ideas that can make a profound difference in how you think about and analyze the world. This is our sweet spot.

In This Course, Thinking Is More Important Than Calculating

The mathematics behind basic statistics is simple (it’s mostly adding, subtracting, multiplying, and dividing a bunch of times). The computations are trivial, though labor intensive. That’s why statisticians, more than anyone, are so excited about the advent of modern computers. It means they don’t have to do repeated and boring calculations. In fact, if all you learn is how to calculate statistics, you really won’t know much.

Statistics is not about calculation. It is about thinking. The hard part of this course will not be understanding all the pieces—the calculations, the symbols, and so on—but understanding how all these pieces fit together. In short, we need to grow our understanding. Let’s pause to consider what it means to understand, and how we can get there.