1 Chapter 1: Introduction to Statistics

Statistical thinking is a way of understanding a complex world by describing it in relatively simple terms that nonetheless capture essential aspects of its structure or function, and that also provide us some idea of how uncertain we are about that knowledge. The foundations of statistical thinking come primarily from mathematics and statistics, but also from computer science, psychology, and other fields of study.

We can distinguish statistical thinking from other forms of thinking that are less likely to describe the world accurately. In particular, human intuition often tries to answer the same questions that we can answer using statistical thinking, but often gets the answer wrong. For example, in recent years most Americans have reported that they think that violent crime was worse compared to the previous year (Pew Research Center). However, a statistical analysis of the actual crime data shows that in fact violent crime has steadily decreased since the 1990’s. Intuition fails us because we rely upon best guesses (which psychologists refer to as heuristics) that can often get it wrong. For example, humans often judge the prevalence of some event (like violent crime) using an availability heuristic – that is, how easily can we think of an example of violent crime. For this reason, our judgments of increasing crime rates may be more reflective of increasing news coverage, in spite of an actual decrease in the rate of crime. Statistical thinking provides us with the tools to more accurately understand the world and overcome the biases of human judgment

Dealing with statistics anxiety

Many people come to their first statistics class with a lot of trepidation and anxiety. Learning statistics, like learning in general, takes knowledgeable teachers, willing students, and, most importantly, a great deal of time and practice. Learning statistics is like learning a language. The symbols and notation make up the rules of grammar and the terminology is the vocabulary. Doing the homework is like practicing the conversation of statistics. Becoming fluent (and staying fluent) in statistics requires practice and continuous use.

Questionnaires can be used to survey students prior to the first class in order to measure their attitude towards statistics, asking them to rate a number of statements on a scale of 1 (strongly disagree) to 7 (strongly agree). One of the items on the statistical attitudes survey is “The thought of being enrolled in a statistics course makes me nervous”. In a recent class, almost two-thirds of the class responded with a five or higher, and about one-fourth of the students said that they strongly agreed with the statement. So if you feel nervous about starting to learn statistics, you are not alone.

Anxiety feels uncomfortable, but psychology tells us that some emotional arousal can actually help us perform better on some tasks, by focusing our attention. So if you start to feel anxious about the material in this book, remind yourself that many other students are feeling similarly, and that this emotional arousal could actually help you learn the material better (even if it doesn’t seem like it!).

Tips for Statistics Anxiety                                                                                                                                                                       1. Learn stress management and relaxation techniques.
Techniques such as deep breathing and meditation that help you to relax in any stressful situation can also be helpful when dealing with the nervousness and tension that affect students with math anxiety.
2. Combat negative thinking.
Lack of confidence can be a major impediment for students with math anxiety. Replace those negative thoughts (“I can’t do this”, “I’ve never been good at math”, “I won’t finish in time”) with confidence-building affirmations (“I know this”, “I’m prepared”, “I can do this”).
3. Visualize yourself succeeding.
Athletes use the technique of “visualization” to prepare for major competitions. Imagine yourself being relaxed doing math and during a test and confidently solving the problems.
4. Do “easiest” problems first.
Build up your confidence by first doing those problems in an assignment or on a test that you “know” best. It’ll help you relax when you tackle the “harder” stuff.
5. Channel your stress into something else.
Free up your mind by relieving some of your physical responses to stress. Get up and run around the hall for a minute before the test or squeeze a stress ball like crazy during the test.
6. Start preparing early.
If you try to “cram” the material quickly, you are likely to forget it quickly too. If you practice the
material over a period of time, you will have a better understanding of it and are less likely to forget it when under stress.
7. Take care of yourself.
Although it’s not easy when you’re in school, eating and sleeping well helps your body and mind function to their fullest potential.
8. Try to understand the “why” of statistical concepts rather than memorizing them.
The first thing to go when you are under stress is your short-term memory. This is one reason it is so important to understand that math is not just a set of rules that you have to memorize but that each concept builds on what came before. If you understand the reason behind the rules, you will remember the concepts better and be able to apply them in many different types of problems (not just ones you’ve seen before).
9. Reward yourself for hard work.
After completing a difficult assignment or an exam, it’s time to give yourself a break.

What can statistics do for us?

There are three major things that we can do with statistics:

1. Describe: The world is complex and we often need to describe it in a simplified way that we can understand.
2. Decide: We often need to make decisions based on data, usually in the face of uncertainty.
3. Predict: We often wish to make predictions about new situations based on our knowledge of previous situations.

Let’s look at an example of these in action, centered on a question that many of us are interested in: How do we decide what’s healthy to eat? There are many different sources of guidance; government dietary guidelines, diet books, and bloggers, just to name a few. Let’s focus in on a specific question: Is saturated fat in our diet a bad thing?

One way that we might answer this question is common sense. If we eat fat, then it’s going to turn straight into fat in our bodies, right? And we have all seen photos of arteries clogged with fat, so eating fat is going to clog our arteries, right?

Another way that we might answer this question is by listening to authority figures. The Dietary Guidelines from the US Food and Drug Administration have as one of their Key Recommendations that “A healthy eating pattern limits saturated fats”. You might hope that these guidelines would be based on good science, and in some cases they are, but as Nina Teicholz outlined in her book “Big Fat Surprise”(Teicholz 2014), this particular recommendation seems to be based more on the longstanding dogma of nutrition researchers than on actual evidence.

Finally, we might look at actual scientific research. Let’s start by looking at a large study called the PURE (Prospective Urban Rural Epidemiology) study, which has examined diets and health outcomes (including death) in more than 135,000 people from 18 different countries. In one of the analyses of this dataset (published in The Lancet in 2017; Dehghan et al. (2017)), the PURE investigators reported an analysis of how intake of various classes of macronutrients (including saturated fats and carbohydrates) was related to the likelihood of dying during the time that people were followed. People were followed for a median of 7.4 years, meaning that half of the people in the study were followed for less and half were followed for more than 7.4 years. Figure 1 plots some of the data from the study (extracted from the paper), showing the relationship between the intake of both saturated fats and carbohydrates and the risk of dying from any cause.

This plot is based on ten numbers. To obtain these numbers, the researchers split the group of 135,335 study participants (which we call the “sample”) into 5 groups (“quintiles”) after ordering them in terms of their intake of either of the nutrients; the first quintile contains the 20% of people with the lowest intake, and the 5th quintile contains the 20% with the highest intake.

The researchers then computed how often people in each of those groups died during the time they were being followed. The figure expresses this in terms of the relative risk of dying in comparison to the lowest quintile: If this number is greater than one, it means that people in the group are more likely to die than are people in the lowest quintile, whereas if it’s less than one, it means that people in the group are less likely to die. Figure 1.1 is pretty clear: People who ate more saturated fat were less likely to die during the study, with the lowest death rate seen for people who were in the fourth quintile (that is, who ate more fat than the lowest 60% but less than the top 20%). The opposite is seen for carbohydrates; the more carbs a person ate, the more likely they were to die during the study. This example shows how we can use statistics to describe a complex dataset in terms of a much simpler set of numbers; if we had to look at the data from each of the study participants at the same time, we would be overloaded with data and it would be hard to see the pattern that emerges when they are described more simply.

The numbers in Figure 1 seem to show that deaths decrease with saturated fat and increase with carbohydrate intake. This large-scale study also had some methodological challenges controlling for socioeconomic factors and measurement of dietary intake data. We also know that there is a lot of uncertainty in the data; there are some people who died early even though they ate a low-carb diet, and, similarly, some people who ate a ton of carbs but lived to a ripe old age. Given this variability, we want to decide whether the relationships that we see in the data are large enough that we wouldn’t expect them to occur randomly if there was not truly a relationship between diet and longevity. Statistics provide us with the tools to make these kinds of decisions. But as we will see throughout the book, this need for black-and-white decisions based on fuzzy evidence can lead researchers astray.

Based on the data we would also like to make predictions about future outcomes. For example, a life insurance company might want to use data about a particular person’s intake of fat and carbohydrate to predict how long they are likely to live. An important aspect of prediction is that it requires us to generalize from the data we already have to some other situation, often in the future; if our conclusions were limited to the specific people in the study at a particular time, then the study would not be very useful. In general, researchers must assume that their particular sample is representative of a larger population, which requires that they obtain the sample in a way that provides an unbiased picture of the population. For example, if the PURE study had recruited all of its participants from religious sects that practice vegetarianism, then we probably wouldn’t want to generalize the results to people who follow different dietary standards.

The big ideas of statistics

Study design is also important part of statistical thinking — remember correlation and causation.  Any introduction to psychology course and introductory statistics will often teach that “correlation does not imply causation”, though the renowned data visualization expert Edward Tufte has added, “but it sure is a hint.”

We will examine more about study design and types of data in our next chapter!

Learning Objectives

1.  Define statistical thinking and why we use statistics.
2. Practice ways to reduce statistical anxiety.
3. Identify how statistical techniques fit into the general process of science.

Exercises – Chapter 1

1. Reflect on a statistic that you have encountered in daily life. How can you apply statistical thinking?
2. What are two reasons that you identified for why taking a course in statistics is important?
3. How would you define statistics to a friend, neighbor, family member? Define statistics from what you have learned so far.
4. Review the tips for statistical anxiety and reflect on how you can implement at least one tip to help you succeed in the course.