Quantitative Insights for Lawyers

1 Basic math 13
Numbers 14
Numbers in context 15
Adding (+) 17
Subtracting (−) 19
Multiplying (x) 20
Dividing 21
To raise to the power (ab) 21
Taking roots 22
Logarithm 24
Calculating with percentages 27
Interest calculation 29
Background material (for those who want to read more) 35
Exercises on Chapter 1 36

2 Basic concepts in statistics 39
Average 41
Median 41
Mode 42
Variance 44
Empirical distribution 45
Compare with theoretical distribution 47
Are there any other distributions? 49
Covariance 50
Correlation 52
Correlation and causality 52
Background material (for those who want to read more) 53
Exercises on Chapter 2 54

3 Simple probability distribution 57
Normal distribution 58
Uniform distribution for discrete observations 60
Uniform distribution for continuous observations 63
Poisson distribution 65
Lognormal distribution 65
Binomial distribution 67
Beta distribution 68
Hypothesis testing, general 70
The p value 72
Pitfalls 74
Background material (for those who want to read more) 75
Exercises on Chapter 3 76

4 Two approaches in statistics 81
Classical statistics 82
On the road to Bayesian statistics 85
Bayesian statistical analysis 87
In legal practice 90
Background material (for those who want to read more) 91
Exercises on Chapter 4 92

5 Simple regression model 95
Extending a line 96
A trend line 100
A simple regression model 100
Step by step 103
How good is the calculated trend line? 104
Measure R2 105
t value 108
How much confidence do we have in a prediction? 108
Background material (for those who want to read more) 110
Exercises on Chapter 5 111

6 Multiple regression model 115
Multiple regression model 116
Differences in salaries between men and women 121
Are the results robust? 125
Background material (for those who want to read more) 127
Exercises on Chapter 6 128

7 Creating and evaluating forecasts 131
Types of forecasts 132
The general idea 133
Conditional and unconditional 135
Predictability and properties of the data 136
Case study: predicting increase or decrease in annual bankruptcies 138
Benchmark 144
Background material (if you want to read more) 144
Exercises on Chapter 7 145

8 Learning from forecast errors 147
Commonly used criteria for evaluation 149
Sources of unusual prediction errors 153
Combining predictions 158
Adjustments by experts 160
Background material (for those who want to read more) 161
Exercises on Chapter 8 162

9 Algorithms 165
K means clustering 168
Random Forest 171
Support Vector Regression 173
Multilayer model 175
Ridge, Lasso and Elastic Net 176
(Artificial) Neural Network 177
Background material (for those who want to read more) 181
Exercises on Chapter 9 182

Answers to selected exercises 185
Examples of multiple-choice questions: test yourself 201