What Is Statistics

# What is data science? #

The term data refers to any collection of observations that measure something of interest, or that convey information about a question at hand. This is a data science course, and also a statistics course. For our purposes, the terms “data science” and “statistics” are essentially synonyms, referring to the methodology used to learn from data. Mathematics and computer science are also important components of data science.

Nearly every branch of science involves collecting and analyzing data, but a “domain scientist” such as a biologist or a sociologist is primarily interested in the core questions of their domain, not in the methods used to analyze data. Data scientists do analyze data, but even more importantly, data scientists analyze the methods for analyzing data. This is what distinguishes a data scientist, or a statistician, from a scientist studying and analyzing a data set that arises in their domain of research.

Good data science, like good statistics, starts with a question. For example, in a business setting, we may have questions about what type of person is most likely to buy a product, or whether people would be willing to pay more for a product that has premium features. In natural and social science, questions are often expressed in the form of a hypothesis. For example, in a medical research setting we may have a hypothesis that “people who sleep less than six hours per night tend to have higher blood pressure than people who sleep more than seven hours per night”. When we express such a hypothesis, we must be open to the possibility that the hypothesis is either true or false. Upon systematically collecting relevant data, we will accumulate evidence that informs us about the truth of our hypothesis.

Data science is part of an empirical approach to answering research questions, meaning that we make progress by observing, taking measurements, and collecting and interpreting data. In contrast, a first principles approach to research aims to answer questions using logical deduction and theory. Logical deduction and theory do play important roles in data science, but in data science we prefer as much as possible to “let the data speak for itself”.

Data science and statistics are “methodological” subjects, meaning that they focus on developing methods, tools, and approaches for conducting empirical investigations. A primary aim of data science is to develop an understanding of the strengths and limitations of various methods for analyzing data. Thus, data science is to some extent a “meta subject” which focuses on the merits of different approaches for learning about reality.

There is a very active theoretical branch of data science that deals with “pure” questions about data analysis that exist outside the context of any specific application. However this course will primarily develop the tools of statistics and data science through case studies that are set in various application domains. There is also a more abstract dimension to this course, because we will see that statistical tools often have properties that hold regardless of the specific type of data or application context in which the tool is applied.

## Uncertainty in data analysis #

Statistical data analysis is based on the idea that the data we collect in order to address our questions of interest can never be sufficient to povide definitive answers. There will always be uncertainty in our findings. The goal of a statistical data analysis is to obtain the strongest conclusions that can legitimately be made from the available data, and then quantify the uncertainty in these findings.

Historically, it has been challenging to formalize exactly what we mean by “uncertainty”. A major advance occurred in the late 1800’s, when probability theory matured as a branch of mathematics. Probability theory turns out to be a very useful tool for defining and quantifying what we mean by “uncertainty”. In spite of decades of progress, there remain many unresolved challenges in statistical data analysis. New methods and approaches to analyzing data continue to be developed, and the strengths and limitations of existing methods continue to be examined. Statistics and data science are dynamic fields, and there is ongoing active discussion and healthy debate as to which approaches to data analysis are most appropriate in various settings.

## Samples and populations #

The most prototypical setting for a statistical analysis is when our data constitute a representative or random sample from a population of interest. We will discuss these terms in much more detail later in the course. For now, we will introduce the main ideas using an example. Suppose that our research goal is to estimate the fraction of adults in the state of Michigan who travel more than 20 miles to work each day. Imagine that we could obtain a representative sample of 1,000 adults in Michigan (which has an adult population of roughly 7.5 million, so our sample contains less than one in seven thousand of the population). If 274 of the people in our sample travel more than 20 miles to work each day, then we would estimate that 274/1000 = 27.4% of the Michigan population travels more than 20 miles to work each day.

The true proportion of Michigan adults who travel more than 20 miles to work each day is very unlikely to be exactly equal to 27.4% (i.e., it is very unlikely that exactly 2.055 million Michigan adults travel more than 20 miles to work each day). Although the true proportion may be quite close to this value, it is very unlikely to be exactly equal to it. The goal of uncertainty quantification is to state how different the estimated proportion obtained from the sample of data that we have collected (27.4%) is from the exact, true proportion (which is unknown).

It turns out that as long as we know some key pieces of information about how our sample was obtained, then it is possible to make precise and useful statements about the likely error in our estimate relative to the truth. On the other hand, if we know very little about how our sample was obtained, it can be very difficult to say anything about such errors. This is a very common theme in statistics and data science – it is very important to understand how the data being analyzed were collected, otherwise we will be very limited in the types of claims that we can make.

## More challenging population settings #

A representative sample from a finite population, as described above, is perhaps the simplest setting in which to conduct a data analysis. Unfortunately, our data and population are usually more challenging to work with.

One such example is “time series” data from a “dynamical system”. Such a system, can be continuously changing, so that the system we observe today differs fundamentally from the system that we observed in the past. Consider, for example, research on the Earth’s climate. We can collect data such as temperature, ice cover, and carbon dioxide levels – but the relationships among these variables may appear to drift over time. It’s not that the laws of nature are changing, but rather there are almost always additional relevant factors beyond what we have measured. As these unobserved quantities change, the relationships among the observable variables may change as well.

Temporal systems with such dynamic behavior arise in many different fields of research. For example, in economics and public policy, there is great interest in the relationship between public debt, unemployment, and inflation. In the past, it was consistently observed that greater government spending was associated with greater public debt, lower unemployment, wage growth, and price inflation. But in recent years, many regions of the world have simultaneously experienced low unemployment, high public debt, and low wage growth, but also low inflation.

A system that is in a constant state of structural change is said to be “nonstationary”. Standard methods for analyzing data from other settings may not give meaningful results here. It is often possible to carry out meaningful empirical research by analyzing data obtained from such systems, but it is very important to be aware that your data analysis is being conducted in such a setting, and to make use of methods that are appropriate for it.

## Causality #

The most interesting scientific findings are usually those that identify causes. For example, a researcher may have a hypothesis that among those with COVID, people who are overweight are more likely to become severely ill. This hypothesis, while interesting, reflects a “predictive” relationship, not necessarily one that is causal or “mechanistic”. For example, it could be that the cause of severe illness in COVID patients is insulin resistance, and overweight people just tend to have insulin resistance (but a non-overweight COVID patient with insulin resistance would be just as likely to become severely ill).

Causal statements are usually much more interesting than statements that are not causal. But causal statements are more difficult to demonstrate. One of the major challenges in data analysis is to identify the situations in which causal conclusions may be drawn. Just as importantly, we should aim to determine when this cannot be done and communicate to our audience that a causal conclusion is not justified.