Introduction

The field of statistic can be very broadly summarized as taking data that we have in hand to make statements that we think have broader generality than just this data.

Consider this data from the Pew Research Center, based on a survey of US Parents in May of 2025 of 3,054 eligible parents who responded (out of 6,287 who were asked)1

This is the results of data collected on a small sample of all US parents, but its interest is lies in being able to use it to infer or make statements about the general population of all US parents.

In statistics this is called inference and is the heart of the subject of statistics. This course is an rigorous, mathematical introduction into the process of inference and inferential thinking about data more broadly.

The main parts of inference can be roughly broken down into the following steps:

  1. How do we use our data to best estimate something about a larger population
    or
    to predict something about future observations we will see from that population?

I estimate 51% of 2-4 yrs old and 54% of 5-12 yrs old in the US watch YouTube daily

  1. How good of an estimate/prediction is it and can we give measures of accuracy or confidence in our estimate/prediction?

With 95% confidence, I estimate 47.9-54.1% of 2-4 yrs old and 50.8-57.0% of 5-12 yrs old in the US watch YouTube daily

  1. How can we make comparisons or test hypotheses from our data?

Are the rates of daily YouTube consumption different between 2-4yr olds and 5-12yr olds in the general population?

Putting all of these together into an analysis of a dataset is the process of inference.

We are able to accomplish these tasks by thinking of our data as random and using probability to model aspects of our data. These probabilistic models are how we justify generalizing beyond our specific data.

You will in the syllabus see that the course topics will progress through these fundamental steps.

Haven’t I already done this in my intro stat course?

If you have taken previous intro statistics courses, like STAT 20 or DATA 8, you have probably been introduced to these topics. That’s great! Most people who take STAT 135 have taken one of these courses. The difference in this course is that we are going to lean heavily into your probability from STAT 134 or 140 to make these ideas mathematically rigorous and go further. The goal is both that you mathematically understand the specific tools we will introduce, and that you have the foundation to expand into more complex scenarios (for example, 150-level Statistics classes!).

There are also things in Stat 20 that we don’t spend a lot of time on, such as visualization tools like boxplots and histograms, or basic summary statistics of data. We assume you’ve seen these basic things at some point.