1 Introduction to Statistics

Learning Outcomes

In this chapter, you will learn how to:

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

What are statistics?

Generally speaking, statistics include numerical facts and figures. For instance:

  • The largest earthquake measured 9.2 on the Richter scale.
  • Men are at least 10 times more likely than women to commit murder.
  • Fewer than 2 percent of NCAA student-athletes go on to the professional level.
  • Over 9 in 10 teenagers ages 13-17 in the U.S. have had two or more doses of the MMR vaccine.
  • The average American generates nearly 4.5 pounds of trash each day.
  • Twenty-one percent of U.S. adults experienced mental illness in 2020.

The study of statistics involves math and relies upon calculations of numbers. But it also relies heavily on how the numbers are chosen and how the statistics are interpreted.

Statistical reasoning involves how numbers are chosen and how statistics are interpreted. Consider the following three scenarios and the interpretations based upon the presented statistics. You will find that the numbers may be right, but the interpretation may be wrong.

Try to identify a major flaw with each interpretation before we describe it.

  1. A new advertisement for Ben and Jerry’s ice cream introduced in late May resulted in a 30% increase in ice cream sales for the following three months. Thus, the advertisement was effective.
  2. The more churches in a city, the more crime there is. Thus, churches lead to crime.
  3. 75% more interracial marriages are occurring this year than 25 years ago. Thus, our society accepts interracial marriages.

What did you come up with?

  1. A new advertisement for Ben and Jerry’s ice cream introduced in late May resulted in a 30% increase in ice cream sales for the following three months. Thus, the advertisement was effective. A major flaw is that ice cream consumption generally increases in the months of June, July, and August regardless of advertisements. This effect is called a history effect and leads people to interpret outcomes as the result of one variable when another variable (in this case, one having to do with the passage of time) is actually responsible.
  2. The more churches in a city, the more crime there is. Thus, churches lead to crime. A major flaw is that both increased churches and increased crime rates can be explained by larger populations. In bigger cities, there are both more churches and more crime. This problem, which we will discuss in more detail later, refers to the third-variable problem. Namely, a third variable can cause both situations; however, people erroneously believe that there is a causal relationship between the two primary variables rather than recognize that a third variable can cause both.
  3. 75% more interracial marriages are occurring this year than 25 years ago. Thus, our society accepts interracial marriages. A major flaw is that we don’t have the information that we need. What is the rate at which marriages are occurring? Suppose only 1% of marriages 25 years ago were interracial and so now 1.75% of marriages are interracial (1.75 is 75% higher than 1). But this latter number is hardly evidence suggesting the acceptability of interracial marriages. In addition, the statistic provided does not rule out the possibility that the number of interracial marriages has seen dramatic fluctuations over the years and this year is not the highest. Again, there is simply not enough information to understand fully the impact of the statistics.

As a whole, the three examples above show that statistics are not only facts and figures; they are something more than that–they are numbers measured for some purpose. In the broadest sense, statistics refers to a range of techniques and procedures for analyzing, interpreting, displaying, and making decisions based on data.

Statistics is the language of science and data. The ability to understand and communicate using statistics enables researchers from different labs, different languages, and different fields to articulate to one another exactly what they have found in their work. It is an objective, precise, and powerful tool in science and in everyday life.

What statistics are not.

Many psychology, social science, education, nursing, and other students dread the idea of taking a statistics course, and more than a few have changed majors upon learning that it is a requirement. That is because many students view statistics as a math class, which is not exactly true. While many of you will not believe this or agree with it, statistics isn’t math. Although math is a central component of it, statistics itself is a broader way of organizing, interpreting, and communicating information in an objective manner. Statistics is a way of viewing reality as it exists around us in a way that we otherwise could not.

Why do we study statistics?

Virtually every student of the sciences takes some form of statistics class. This is because statistics is how we communicate in science. It serves as the link between a research idea and actionable conclusions. Without statistics, we would be unable to interpret the massive amounts of information contained in data. Even small datasets contain hundreds – if not thousands – of numbers, each representing a specific observation we made. Without a way to organize these numbers into a more interpretable form, we would be lost, having wasted the time and money of our participants, ourselves, and the communities we serve.

Beyond its use in science, however, there is a more personal reason to study statistics. Like most people, you probably feel that it is important to “take control of your life.” But what does this mean? Partly, it means being able to properly evaluate the data and claims that bombard you every day. If you cannot distinguish good from faulty reasoning, then you are vulnerable to manipulation and to decisions that are not in your best interest. Statistics provides tools that you need in order to react intelligently to information you hear or read. In this sense, statistics is one of the most important things that you can study.

To be more specific, here are some claims that we have heard on several occasions. (We are not saying that any of these claims are true!)

  • Almost 85% of lung cancers in men and 45% in women are tobacco-related.
  • Condoms are effective 94% of the time.
  • People tend to be more persuasive when they look others directly in the eye and speak loudly and quickly.
  • Women make 75 cents to every dollar a man makes when they work the same job.
  • Using Facebook lowers your grades.
  • There is an 80% chance that in a room full of 30 people that at least two people will share the same birthday.
  • 67.53% of all statistics are made up on the spot.

All of these claims are statistical in character. We suspect that some of them sound familiar; if not, we bet that you have heard other claims like them. Notice how diverse the examples are. Indeed, data and data interpretation show up in discourse from virtually every facet of contemporary life.

Statistics are often presented in an effort to add credibility to an argument or advice. You can see this by paying attention to advertisements. Many of the numbers thrown about in this way do not represent careful statistical analysis. They can be misleading and push you into decisions that you might find cause to regret. For these reasons, learning about statistics is a long step towards taking control of your life. (It is not, of course, the only step needed for this purpose.) The purpose of this course is to help you learn statistical essentials and help prepare you for a career in psychology, nursing, counseling, physical therapy, occupational therapy, social work, education, or other fields that use evidence-based decision making. Again, taking this course will make you into an intelligent consumer of statistical claims.

You can take the first step right away. To be an intelligent consumer of statistics, your first reflex must be to question the statistics that you encounter. The British Prime Minister Benjamin Disraeli is quoted by Mark Twain as having said, “There are three kinds of lies — lies, damned lies, and statistics.” This quote reminds us why it is so important to understand statistics. So let us invite you to reform your statistical habits from now on. No longer will you blindly accept numbers or findings. Instead, you will begin to think about the numbers, their sources, and most importantly, the procedures used to generate them.

The above section puts an emphasis on defending ourselves against fraudulent claims wrapped up as statistics, but let us look at a more positive note. Just as important as detecting the deceptive use of statistics is the appreciation of the proper use of statistics. You must also learn to recognize statistical evidence that supports a stated conclusion. Statistics are all around you, sometimes used well, sometimes not. We must learn how to distinguish the two cases. In doing so, statistics might be the course you use most in your day to day life, even if you do not ever run a formal analysis again… you will use statistical thinking!

What is statistical thinking?

“Statistical thinking will one day be as necessary for efficient citizenship as the ability to read and write.” – H.G. Wells

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

What can statistics do for us?

Let’s say we’re interested in answering this 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, in which researchers 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.  They found that people who ate more saturated fat were less likely to die during the study, with the lowest death rate seen for people who who ate more fat than the lowest 60% but less than the top 20%. The opposite was seen for carbohydrates; the more carbs a person ate, the more likely they were to die during the study.

There are three major goals for 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.

When answering the question using scientific research, the preferred method, the researchers were able to describe a complex dataset in terms of a much simpler set of numbers; if they had to look at the data from each of the study participants at the same time, they would have been overloaded with data and it would have been hard to see the pattern that emerged.

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.

Based on the data we may 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.

The big ideas of statistics

One way to think of statistics is as a set of tools that enable us to learn from data.

There are two main branches of statistical analysis, descriptive statistics and inferential statistics.

  • Statistics that are used to organize and summarize the information so that the researcher can see what happened during the research study and can also communicate the results to others are called descriptive statistics. The first unit of the book is focused on descriptive statistics.
  • Statistics that help the researcher to answer a research question by determining exactly what conclusions are justified based on the results that were obtained are referred to as inferential statistics.

Statistics provides us with the tools to characterize uncertainty, to make decisions under uncertainty, and to make predictions whose uncertainty we can quantify. For example, we now know that cigarette smoking causes lung cancer, but this causation is probabilistic: A 68-year-old man who smoked two packs a day for the past 50 years and continues to smoke has a 15% (1 out of 7) risk of getting lung cancer, which is much higher than the chance of lung cancer in a nonsmoker. However, it also means that there will be many people who smoke their entire lives and never get lung cancer.

One often sees journalists write that scientific researchers have “proven” some hypothesis. But statistical analysis can never “prove” a hypothesis, in the sense of demonstrating that it must be true (as one would in a logical or mathematical proof). Statistics can provide us with evidence, but it’s always tentative and subject to the uncertainty that is always present in the real world.

Study design is an important part of statistical thinking.  We will examine more about study design and types of data in our next chapter.

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 instructors, 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 have been 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!).  As Dr. Kelly McGonigal says, you should “Make Stress your Friend“.

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”), or more growth mindset thoughts (“I may not be able to do this yet, so I’ll keep practicing until I can”).
3. Visualize yourself succeeding.
Athletes use the technique of “visualization” to prepare for major competitions. Imagine yourself being relaxed doing math and confidently solving the problems.
4. Do “easiest” problems first.
Build up your confidence by first doing those problems 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 or squeeze a stress ball like crazy.
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 task or problem, it’s time to give yourself a break… or maybe a cookie.

 

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Introduction to Statistics for the Social Sciences Copyright © 2021 by Jennifer Ivie; Alicia MacKay is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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