The journey to master statistics starts with a simple concept - uncertainty. We all know what it feels like to be uncertain. Some think of it as a gut feeling. Many books begin with the mundane details such as data types or programming language basics; we offer those details here. We also offer structure to better understand what statistics do for us. Statistics is a tool designed by humans to serve humans. If you are a human and vote either with your wallet and/or at the ballot box, you might find this book helpful. Voting is a decision that ought to be guided by your uncertainty. Putting numbers to your uncertainty is the aim of statistics. We will guide you through the learning process.
Should I read this book?
We have no clue if you should read the book; only you can answer that question. Check these hints for yourself:
If you answered yes to any of these questions, you might find this book useful. We hope you read on.
Not a graduate student? That is fine. Read the book anyway. Arm yourself with knowledge of how uncertainty gets communicated. You - the reader - can make more informed decisions when you quantify your uncertainty. We wrote this book with graduate students in mind since most had a previous undergraduate course in statistics that, no doubt, was taught a very standard way. That way does not bode well for long-term retention. Most students forget the details and only a small minority retain the essentials. We assume only a passing familiarity. Most people have the background we assume in this book.
Statistics is a tool to do one thing: quantify uncertainty. Approach the topic with that frame of reference and you will master statistics. How quickly? Depends upon your willingness to devote time to read and apply what you learn. We offer you a path towards understanding, quantifying and mastering your uncertainty. Good luck and welcome to the show.
What We Expect from You
As implied in our self-check above, you need to read English, be motivated and have some resources to both read this book and run the code we provide. You must be willing to read and apply what you learn.
What To Expect
You should expect:
Structure: A framework to understand and master statistics
Brevity: short, concise descriptions of key statistical concepts
Examples: real-world examples to illustrate the concepts
Code: statistical software code to demonstrate the concepts
Challenges: exercises to test your understanding of the concepts
References: a list of references to deepen your understanding of the concepts
How To Read This Book
Pace yourself. Expect familiarity after your first read, some facility after your second read, and moderate mastery after your third read. So, expect to read the book 3 times. Why? We know that repetition is key to mastery. Lather, rinse, repeat. If it is good enough for your hair, it surely is good enough for your brain.
Mastery of Statistics
My (PEM) graduate advisor Lee Sechrest often likened learning to a helix. Every student enters the helix at the same level (bottom) when they begin learning any topic and spiral upward. Some students spiral faster than others; some students start higher than others. The goal is to spiral upward - a motion towards mastery. Will you master statistics in one semester or one resource? No. Mastery is aspiration. Keep pushing.
The Helix Model of Learning
Click me to peek at the code!
# Load the necessary librarylibrary(plotly)# Create data for the helix: every learner enters at the bottom and spirals# upward toward mastery.theta <-seq(0, 6* pi, length.out =500)helix_df <-data.frame(x =cos(theta),y =sin(theta),z = theta,lvl =rep(c("1. Basic Arithmetic", "2. Simple Calculations","3. Conceptual Understanding", "4. Principled Application","5. Mastery"), each =100),mperc =seq(0, 100, length.out =500))# Precompute a plain-text hover label per point. (Referencing non-existent# plotly fields with format codes like %{lvl:.0%} makes the 3D plot fail to# draw; a ready-made text label with hoverinfo = "text" is the robust idiom.)helix_df$label <-paste0(helix_df$lvl, "<br>Mastery: ", round(helix_df$mperc), "%")# Create the 3D helix. It sits on a light figure card (see brand.scss), so the# default light background + dark text read cleanly on either book theme.plot_ly(helix_df,x =~x, y =~y, z =~z,type ="scatter3d",mode ="lines",color =~lvl,line =list(width =6),text =~label,hoverinfo ="text") %>%layout(legend =list(title =list(text ="Level of Mastery")),scene =list(xaxis =list(title ="", showgrid =FALSE, zeroline =FALSE, showline =FALSE, showticklabels =FALSE),yaxis =list(title ="", showgrid =FALSE, zeroline =FALSE, showline =FALSE, showticklabels =FALSE),zaxis =list(title ="", showgrid =FALSE, zeroline =FALSE, showline =FALSE, showticklabels =FALSE) ) )
Figure 1: The Helix Model of Learning
A 30,000 ft View of Statistics
The structure begins here. We offer you a 30,000 ft view of statistics and guide you through an example, highlight some key concepts, define them, and then demonstrate their usefulness in the entire endeavor. Our framework allows you to revisit these key terms and concepts as you progress through the book.
Uncertainty
To begin, the structural core of statistics is uncertainty. We know not what we don’t know but we can quantify it with (false) precision. Quantifying uncertainty is the singular aim of all statistical methods. Methods include estimation, summary, prediction, and testing. Ultimately, people who use statistics either serve or serve as scientists. Our primary aim is to ensure that our scientists can use these tools effectively to quantify their own uncertainty. We share with you (the world) the same materials.
The Scientific Method (for reals)
We learn early on in school a serious falsehood. Science is not a lock-step path of hypothesis generation, stringent testing, and careful reporting. Far from it. There are scientists and scoundrels among us. Most of us try to do the best we can with what we have available. Some fabricate data. Some fabricate studies. Most of us do our best to conduct research consistent with our scientific mission - to describe, predict, and control the world around us. That sounds so sinister but those are the aims of science. Instead of the lock-step scientific method, we often come up with questions that are routinely rejected (\(p > 0.05\)) by data we scarcely know much about regarding the reliability or, gasp, the validity of our measures. After reading this book, you will learn the background to why those statements are so easy to defend in most scientific areas.
The real scientific method as practiced involves hundreds of hours reading relevant literature, searching for testable assumptions (aka “holes in the literature”), designing studies, cleaning/analyzing data, and interpreting the results. All of these steps often take place with limited resources. We must make the most of what we have available. Social scientists often study unobservable constructs (e.g., intelligence, personality, etc.) and must rely on observable indicators (e.g., test scores, self-report measures, etc.) to make inferences. The inferential process is fraught with uncertainty.
We must quantify that uncertainty to make the best decisions possible. We must also be honest about the limitations of our data, our methods, our inferences, and our conclusions. We must be honest about the limits of our knowledge and our understanding.
That house of cards leads to many researchers hunting for new, potential findings worthy of further inquiry. Failed tests lead to fishing expeditions to find something worthy of publication. Why? We must publish to keep our jobs. So, we find stuff to publish. Failed initial tests? Don’t despair! You will find something to report in every dataset. Sir Ronald Fisher, interpreted the results of significant hypothesis tests worthy of further study - nothing more. He too opined that such tests were mere reason for optimism and not the ultimate arbiter of scientific relevance. Thus, all findings are findings worthy of further study. Sometimes, that further study stops with greater care and attention to the data; other times, we must conduct further empirical investigations to rule out alternative explanations. Here and further along in your journey, you will come back to this first paragraph and (hopefully) reflect on your own uncertainty.
EXAMPLE 1: Ice Cream Preference
Imagine a boring world where everyone liked chocolate ice cream. We would have no need for statistics or science for prediction; we have both perfect prediction and certainty. If just one person liked vanilla instead of chocolate, we would have uncertainty - not much uncertainty because it would still be prudent to guess that everyone liked chocolate ice cream. One error that occurs 1/N times can be quite trivial with a large sample. This seemingly trivial example brings all of the statistics you need to undestand in English. Now, we need to take these ideas from English and convert them into statistical language. Don’t worry, the language will become second nature and your understanding will deepen with each passing use. We introduced many concepts in this first example including:
ImportantVariables (and constants)
A variable is a characteristic that can take on different values. In this case, the variable is the ice cream flavor that a person would order. The variable can take on different values such as chocolate, vanilla, strawberry, mint chip, and rocky road. A constant is a value that does not change. In this case, the constant is the number of people who like chocolate ice cream. Thus, you now know that when we say variable, we mean some characteristic that we may want to better understand (or use to predict something that we want to understand - more on that later). When we say constant, we mean a value that does not change. Variables vary; constants do not.
Some Examples
Variables: height, weight, age
Constants: \(\pi\), \(e\), \(c\)
Some Demonstrations
# create a vector of heightsheights <-c(5.5, 6.0, 5.8, 5.9, 6.1, 5.7, 5.6, 5.9, 6.2, 5.8)# create a vector of weightsweights <-c(125, 189, 220, 175, 145, 147, 256, 127, 155, 184)# plot the heights using ggplotlibrary(ggplot2)ggplot(data.frame(heights), aes(x = heights)) +geom_histogram()
`stat_bin()` using `bins = 30`. Pick better value `binwidth`.
ImportantVARIANCE!
Yes, we yelled it. If you learn nothing else from this resource, please master VARIANCE. As we alluded to above, the characteristics that we measure differ between people (e.g., height) and within a person (e.g., weight). To our example, we can imagine ice cream preference differing between people (i.e., some people prefer vanilla to chocolate) and within each (i.e., our tastes change as we age). The differences between or changes within are variance. You will learn that there are formal definitions, graphical depictions, and qualifiers to variance. For now, remember that variance is a measure of how much difference exists in your variables. More differences are preferable to fewer differences. Why? Differences are what we study. We study differences to understand the world around us, to predict the future, and to gain some control over the world. Variance is the measure of differences and, as a result, variance is the measure of uncertainty. We shall return to this concept often.
ImportantPrediction
We aim to predict in science.
ImportantError (difference between observed and predicted value)
Error is the gap between what we observed and what we predicted: \(e = \text{observed} - \text{predicted}\). Do not read “error” as “mistake.” It is the part of the world our prediction failed to capture, and working to shrink it and understand it is most of what statistics actually does.
ImportantProbability (likelihood of a particular event occurring)
Probability puts a number on how likely an event is, scaled from 0 (never happens) to 1 (certain). It is the common scale we use to express uncertainty. Every claim we make later, every p-value and every confidence interval, is expressed as a probability.
ImportantFrequency (number of times an event occurs)
Frequency is simply how often something happens - a raw count. It is the most honest number in statistics because you can literally tally it. Pile those counts up across every value a variable can take, and you have built a distribution.
ImportantDistribution (pattern of frequencies of events)
A distribution is the pattern those frequencies make - the shape of how often each value shows up. Almost everything we do from here on is, at bottom, a claim about a distribution: its center, its spread, and how surprising a particular value is within it.
ImportantSample (subset of a population)
A sample is the subset of the population we actually get to observe. We almost never see the whole population (every voter, every patient, every ice-cream eater); the sample is our narrow window onto it, and the entire enterprise of inference is guessing about the population through that window.
ImportantSample size (number of observations in a sample)
Sample size (\(N\)) is the number of observations in our sample. Bigger samples buy us more precision - less uncertainty about the population - which is why \(N\) turns up in nearly every formula in this book. More is not always better, but it is almost always more certain.
Concepts such as variables (and constants), prediction, and sample size. We also introduced the concept of error. Error is the difference between the observed (aka true) value and the predicted value. In this case, the true value is the ice cream flavor the person would order. The predicted value is the most likely flavor the person would order. The error is the difference between the true value and the predicted value. In this case, the error is the difference between the flavor the person would order and the flavor we predicted the person would order.
# Start with a small, manageable sample sizeN <-100# create a vector of ice cream flavorsflavors <-c("chocolate", "vanilla", "strawberry", "mint chip", "rocky road")# create a data frame of ice cream preferencesic_prefs <-data.frame(id=1:N, flavor =factor(rep("chocolate", N), levels = flavors))# create a boxplot of the chocolate sample using ice cream cones with the color of the flavor as the filllibrary(ggplot2)ggplot(ic_prefs, aes(x = flavor)) +geom_bar(fill ="chocolate") +theme_minimal()
# create a sample of 1000 people who like chocolate ice cream 90% of the timesample <-sample(flavors, 1000, replace =TRUE, prob =c(0.95, 0.01, 0.01, 0.01, 0.01))# create a table of the sampletable(sample)
# create a sample of 1000 people who like chocolate ice cream 50% of the timesample <-sample(flavors, 1000, replace =TRUE, prob =c(0.5, 0.1, 0.1, 0.1, 0.2))# create a table of the sampletable(sample)
Who might be such a person to go against the social norms?
one of the early contributors to frequentists statistics, once said, “To consult the statistician after an experiment is finished is often merely to ask him to conduct a post mortem examination. He can perhaps say what the experiment died of.” Such a post mortem were apropos.
Each require some must start with the first general principle - uncertainty or rather quantifying uncertainty. To fully grasp the concept of uncertainty, we offer you a simple exercise. Suppose you were to guess the weight of any adult family member. Your best guess would likely be off. Empirical evidence, however, suggests that the best guess is the arithmetic mean. Why?
[1] 172.3
Min. 1st Qu. Median Mean 3rd Qu. Max.
125.0 145.5 165.0 172.3 187.8 256.0
