In this section of the website, you'll find examples and exercises that we've used for the course, more-or-less organized by date. Sooner or later they'll also be linked into the syllabus by week.

read more# Contents:

# Lessons

In this section of the website, you'll find all of the lessons written for this course. They'll (soon) be organized more or less by date, as soon as I re-date them. Right now, you can just click through the below. There's also a tags page which you can use to …

read more# Problem Sets

In this section of the website, you'll find the current problem sets and, after the due date, answers to them.

read more# Archived Syllabus (2019)

**This is an archived version of the syllabus from the first iteration of the course.**I've left it here for the purposes of maintaining a record.# Introduction to Quantitative & Computational Legal Reasoning (LAW:8645)

Spring 2019; Monday + Tuesday 12:40-2:10; Classroom 125.

Professor: Paul Gowder (Office: 408. Email: paul-gowder …

read more# Post-Week 1 Notes

Well, we've survived our first week of Sociological Gobbledygook (and one room change---which we're keeping, BTW). We've learned the basic mechanics of programming, and, more importantly, we've worked on building a mental model for how talking to a computer works by completing a number of exercises.

For the in-class time …

read more# Problem Set 1

This problem set is worth 17.5% of the grade in this course. It is due on Friday, February 1, at 5pm, via a dropbox in ICON that I shall set up for the purpose.

There are three problems on this set, one relatively small/easy one worth 20 points …

read more# Simulation for fun and profit

One really useful thing that you can do with programming skills is simulate things that you don't fully understand. This is something that hard science people do all the time: it turns out that there are some kinds of problems that aren't solvable in a deductive, analytic kind of way …

read more# Problem Set 2

Due Friday, February 22, at 5pm Central time.

As before, please turn in your answers in a single notebook. For the questions that require you to write down prose, hopefully by now you've realized that you can change the type of a notebook cell to "markdown" to write regular text …

read more# Pset 1 answers

# Week 4 recap: Total Probability Rule

The probability lecture this week kind of hit a wall at one point in our Bayes Rule example. Here's a clearer explanation.

Remember, we had an example problem involving figuring out the posterior probability of someone being drunk, given that they blew a positive result on a breathalyzer.

The place …

read more# Problem Set 3

Due Friday, March 29, at 5pm Central time.

As before, please turn in your answers in a single notebook, emailed to Diana Dewalle.

## Problem 1 (55 Points): Legislative Discrimination

Go onto google scholar and find Daniel M. Butler & David E. Broockman, "Do Politicians Racially Discriminate Against Constituents? A Field Experiment …

read more# Using libraries in Python

Programming would be incredibly time consuming if you couldn't reuse code. You've seen the basic form of code reuse already---the function, which allows you to lock up a transformation from some data to some other data, give it a name, and then apply it repeatedly, and to arbitrary data.

Well …

read more# Introduction to Python and setup.

# Why Statistics for Lawyers?

The brunt of this course will be devoted to statistics and exploratory data analysis.

read more*Exploratory data analysis*is just looking at data to see what you see. We will spend some time, for example, looking at how to see the shape of data and what that can tell you about …# Introduction to Programming and Python

In this first week of the course, we're going to cover some core concepts of computer programming. With this, we will build a foundation to do more interesting things in the coming weeks.

## What is Computer Programming?¶

Computers basically just do one thing: they complete very simple operations at astonishing speed.

For example, suppose you wanted to calculate the thousandth number of the Fibonacci Sequence read more

# Introduction to Statistical Power

Statistical power is, in terms of practical use, kind of the flip side of a p-value.

Remember that a p-value is an attempt to quantify our degree of confidence in not having made a type I error (although it is

read more*not*the probability of having made a type I error …# Introduction to Exploring Data in Python

# Introduction to Exploring Data in Python¶

In this lesson, we're going to learn about how to get a feel for data in Python, using basic tools to look at our data.

First, however, let's make sure we have the right version of a library we need called Seaborn. Run the code in the next block and make sure that you get the right version.

read more# Files and How Computers Represent Data

In this lesson, we're going to learn how to open files and work with data from the disk. We'll start with the mechanical process of opening text files, and then move on to learn a little bit more about different kinds of data you'll see.

Here's the basic method of opening and reading text files. Suppose I have a file called hello.txt in my working directory. (Your working directory is the directory you run Python from on your hard drive. For those of you using Azure Notebooks, this should be your library, but talk to me if you see a file there and can't read it from Python.)

read more# Introduction to Distributions

## What's a Distribution, Anyway?¶

Statistics people often talk about distributions, like a normal distribution. Here's what they mean: suppose you could see all of the instances of the thing you're trying to study. What kind of pattern would their values have? That's the distribution.

For example, suppose you expect that most of the values of the thing you care about will be clustered around some average value. IQ is a good example: most IQs in the population are around 100, and then as values get further away from 100 in either direction, the fraction of the total number of instances that takes that range of values gets smaller. There are lots of folks with an IQ between 85 and 115, fewer between 70 and 85 on one side, and 115 and 130, many fewer between 55 and 70 or 130 and 145, and a (proportionally) truly tiny number between 40-55 or 145-160.

read more# The Normal Distribution and the Central Limit Theorem

The main reason scientists like the normal distribution so much is because of a two little ideas called

**the law of large numbers**(LLN) and the**central limit theorem**(CLT).I'm not going to walk you through proofs of these; instead, we'll just look at some graphs and talk about some intuition.

read more# When Regressions Attack

This lesson is all about what can go wrong in linear regression. Here's an outline of the ways things can go wrong.

- data isn't linear
- extreme outliers

- heteroskadiscity
- multicolinnearity
- conditioning on a collider
- counfounder bias
- non-normal residuals

There's also a problem known as "autocorrelation" which mainly appears in time series data (i.e., when one tries to run a regression on something that changes over time, like stock market prices). Time series analysis is a fairly advanced topic that is beyond the scope of this course, but you should have alarm bells ringing if anyone tries to do ordinary linear regression on data that spans time like that.

read more- data isn't linear
# Getting Data from the Internet With Python

In addition to reading files locally, you can also read them over the internet.

When you use a web browser like Chrome to go to a URL ("uniform resource locator," or web address) like https://sociologicalgobbledygook.com, what you're actually doing is sending a request using the HTTPS protocol (which …

read more# The Basics of Probability

## What is Probability?

Probability is the mathematical representation of the likelihood of an event under a given set of circumstances (conditions) in a given period of time. We will say, for example, that the probability of winning the jackpot in the lottery from buying one ticket this week is some …

read more# Hypothesis Testing: Conceptual Introduction (draft)

Now that we understand distributions and the central limit theorem, we’re in a good position to make sense of the notion of a hypothesis test. It’s actually very simple.

Suppose you do an experiment. Let’s say you want to find out whether a company is engaging in …

read more# Abel and Baker Redux: Probability and Description

Right at the beginning of the F&L readings for this week, you saw a well-known probability puzzler/"paradox." I'm going to risk angering the copyright gods and quote it in full here:

Assume that boys and girls are born with equal frequency. Mr. Able says, “I have two children …

# P-Values and Bayes Rule

Recall from the previous lesson what a p-value is: it’s the probability of observing a value of your statistic as extreme (as far away from the null hypothesis statistic) as you in fact observed, if the null hypothesis were true.

In other words, if you’re doing a (two-sided …

read more# Some (Edited) Cases on Discrimination Testers

Here are three cases illustrating the use of discrimination testers in real-world litigation contexts. If this is hard to read, you can download a pdf version instead.

## United States v. Balistrieri, 981 F.2d 916

1992-11-24

MANION, Circuit Judge.

The United States sued Joseph Balistrieri, owner of the Shorecrest Apartments …

read more# Functions and Scope

Recall how in the first Python lesson we looked at the

`while`

loop and saw how it allows us to repeat instructions to the computer as many times as you want.The next step up from a loop is a function, which allows us to wrap up a series of commands into a single command on its own. Let's take a look at an example.

read more# Common Data Transformations

It's often useful in performing data analysis to transform some of your variables to fit a common scale; this is especially useful in exploratory data analysis, because these transformations often make it much easier to eyeball the relationship between variables. (Also, some statistical techniques require these transformations.)

In this short lesson, we'll introduce two common methods of transforming data---the

*log transform*read more# Introduction to Linear Regression

The standard technique for measuring the relationship between one or more continuous independent variables and a continuous dependent variable is linear regression.

The basic idea of linear regression can be expressed simply. A linear regression is a line (or some more dimensional geometric thingy) that maps the independent variables to the best predicted value for the dependent variable.

read more# More Loops and Control Flow

In this lesson, we'll think about more ways to direct Python to do things repeatedly, or conditionally.

Let's start with more loops. I showed you the while loop before, remember?

# Simple Data Types (draft)

In Python, the data you work with (like the things assigned to variables) have

*types*, which specify the kinds of data they are and the things you can do with them.A good way to understand this is to think about the difference between letters and numbers. While we can write both down, there are different things we can do to them. It wouldn't make sense (except in an algebra context) to multiply and divide letters; it would't make sense to talk about a capital and a lowercase number 3.

read more# Complex Data Types

# Key Python Libraries for Working with Data

In this lesson I'm just going to describe the main libraries that we'll see when we work with data in Python.

## Numpy¶

Numpy is the first library we work with. By convention, it's imported with

`import numpy as np`

. Numpy really provides two things to our workflow:Math that goes faster than unadorned Python could do it---which is important when you're doing statistics, because under the hood computational stats can take a lot of calculations.

read more

# Practical Basic Hypothesis Tests

In this lesson, we're going to very quickly rip through the basic hypothesis tests, their uses, and how to achieve them in Python. I won't spend a lot of time on this, because the mathematical details are covered in the assigned reading, and, at any rate, I think for practical purposes regression analysis is more important for lawyers. Also, this is basically AP/undergrad stats material, so you've probably seen it somewhere already.

read more# Dealing with Errors

An unavoidable fact of life for people who write code is error messages. You're happily programming along, and then, all of a sudden, you get a massive screen of terrifying text telling you that

*you screwed up!!*# Object-Oriented Programming

read more**Object-oriented programming**(OOP) isn't all that special, it's just a particular style of programming that Python is particularly well designed for. This is a short lesson, we won't cover the theory of OOP, or features you might hear about elsewhere like "inheritance"---see your reading in the P4E book for more.# Regular Expressions

Regular expressions (or "regex"/"regexes") are one of the most powerful programming tools for lawyers. Essentially, regular expressions are a powerful specialized programming language built into other languages like Python, which allow you to express complicated text searching operations.

The utility of this for lawyers should be obvious: lawyers have to deal with lots and lots and lots of documents, and sometimes need to search through those documents for specific information. If those documents are in electronic form, regular expressions can provide you with a much more powerful way of searching than what is built into ordinary applications.

read more# In-class example: talking to an API

This is a lightly edited version of the notebook that we worked through in class on 1/22/19.

In class, we went through how to make an API call end-to-end, to get a look at common tasks like figuring out documentation, using libraries, making HTTP requests, etc. Over the weekend, practice with this API and others (you might also try the one at opensecrets.org read more

# In-Class Exercise: Data Scavenger Hunt

## Pandas Filtering

Before we do our scavenger hunt, here's a piece of useful information you should know: you can filter rows of a Pandas dataframe using something that returns a boolean.

For example, suppose you have a DataFrame called copdf with police officer names in a column called "name" and …

read more# Week 5 Recap

In week 5, we began by continuing our probability lecture from last week, and then, as an exercise, tried to prove the correct answer to the Monty Hall problem using Bayes Rule.

## Monty Redux

Here's that solution again. Remember our formula for Bayes Rule:

$$P(B|A) = \frac{P(A …read more# Causation and Counterfactuals

In law as in science, one thing we typically want to resolve are questions of causation. For example, in biology we might ask things like "does this drug reduce deaths from heart disease," in economics we might ask "does raising the minimum wage increase unemploment." In law, such questions will …

read more# Solutions to Data Scavenger Hunt

Here are some example ways to respond to the prompts in our data scavenger hunt.

# Hypothesis Tests on Experimental Data: Housing Discrimination Test Example

# Pset2 answers

# Shonubi/Bootstrap exercise

# Post-Week 9 Notes

Two notes following week 9 (our intro to regressions).

read more**First**, a couple students asked whether you need to center and scale (standardize) data to make it work with linear regression. I think I had a brain freeze and gave an inconsistent answer, so let me clarify here: no, you don't …# Simpson's Paradox Homework Example

Here is a combination of some of the code Sam kindly showed us in class, plus the visualizations I showed you, for our simpson's paradox example on 3/25/19.

Here are a few additional FYIs:

The source of the underlying dataset is an article entitled "Simpson’s Paradox: A Data Set and Discrimination Case Study read more

# Last Problem Set! (Pset 4)

Due Thursday, May 9, at 5pm Central time. This is an

**absolutely firm**deadline, as it is the last day of the exam period.As before, please turn in your answers in a single notebook, emailed to Diana Dewalle.

## "Problem" (not really a problem) 1: Help Next Year's Students (15 …

read more# Makeup Assignment: Research Design

In lieu of the makeup session for the class we missed---which, it turns out, conflicts with every schedule under the sun---please complete the following short assignment. This assignment will be graded on a pass-fail basis, based on effort (moderate effort, not extreme effort---I don't want this to take more than …

read more# Confidence Intervals and Bayesian Statistics oh my!

One of the readings for week 13, "The Bayesian New Statistics," covers a variety of different approaches to statistics, as contrasted with the standard frequentist hypothesis-testing method. I don't expect you to come out of this class being able to work any of those alternative paradigms, but you should be able to recognize them and understand broadly how they operate. That article is a very good summary of the landscape, but this supplemental lesson aims to provide a briefer and slightly more basic introduction.

read more# Prediction vs. Inference

So far, our statistics material in this course has fallen into two buckets. The first, and most straightforward, is

read more*descriptive statistics*, that is, just describing what our data looks like---mean, median, correlation, that kind of stuff. The second is "inferential statistics," that is, use of statistics to make inferences about …# Problem Set 1, Spring 2020

This problem set is worth 17.5% of the grade in this course. It is due on Friday, February 7, at 5pm, via email to Diana Dewalle. It has four short questions, each of which is worth 25 points.

read more**How to turn in this problem set**: each of the problems …# Problem Set 2, Spring 2020

This problem set is worth 17.5% of the grade in this course. It is due on Friday, March 6, at 5pm, via email to Diana Dewalle.

read more**How to turn in this problem set**: As before, please turn in all your code, and, if you have any prose to write …# Day 1 in-class examples (2020)

# In class-examples, January 27, 2020

# In class-examples, January 28, 2020

# In class-examples, February 3, 2020

# In class-examples, February 4, 2020

# In class-examples, February 10, 2020

# In class-examples, February 11, 2020

# In class-examples, February 17, 2020

# Problem Set 1: Answers and Explanations

## Problem 1: Your First Class¶

Write a class, called

read more`Citation`

, which takes the following required parameters:`first_page`

(an integer),`last_page`

(an integer),`reporter`

(a string), and`year`

(an integer), as well as the optional parameter`name`

(a string). That class should have the method`cite()`

which will print out a citation to the case, using all of the information it has.# In class-examples, February 25, 2020

# In-Class Notebook, Mar 2, 2020

# Installing an external library on the UI systems

It's a bit difficult to install libraries to make them usable with jupyter notebooks in general; it's slightly more difficult to make it work in a permission-controlled environment. However, if you want to try my plottyprint library for problem set 2 (which has the advantage of being easy to use), here's some code to try.

read more# In-Class Notebook, Mar 3, 2020

There are a bunch of different ways we might think about the example of the application from the fake data from yesterday. We saw a test where the null hypothesis was that the application offering rate for black renters is equal to the overall application offering rate. With that hypothesis, we saw a binomial test, and here's a slightly more filled out version of that test.

read more# Consolidated Problem Sets 3 and 4, Spring 2020

This problem set is worth 55% of the grade in this course. It is due on Friday, May 8, at 5pm, via email to Diana Dewalle.

read more**How to turn in this problem set**: As before, please turn in all your code, and, if you have any prose to write, your …# Problem Set 2: Answers and Explanations

## Problem 1: Fun with APIs, continued (30 points)¶

Remember problem 3 from the previous pset? I'd like you to go back and use the Caselaw Access Project API again, only, this time, I'd like you to plot line charts of the following two time series, on the same chart:

The total number of the uses of the words "pork," "pig," "pigs," "hog," or "hogs" in the Iowa state courts within the CAP dataset, and

read more

# Problem Set 3, Spring 2020

This problem set is worth 25% of the grade in this course. It is due on Friday, April 3, at 5pm, via email to Diana Dewalle.

read more**How to turn in this problem set**: As before, please turn in all your code, and, if you have any prose to write, your …# Some Rules of Thumb for Controlling

You've seen a lot of talk about controlling for things, for e.g., in multiple regression (or more generally, conditioning on things). It's worth having a quick list of some rules of thumb that will suffice for many cases. Of course, the design of observational studies is much more complicated …

read more