Appendix B: Statistical Tables and Soccer Benchmarks

This appendix collects reference tables for the probability distributions, critical values, and soccer-specific benchmarks used throughout the text. While modern software eliminates the need to look up values by hand, these tables provide useful reference points for quick estimation and sanity-checking results.

B.1 Probability Distributions

B.1.1 Standard Normal Distribution

The standard normal distribution $Z \sim \mathcal{N}(0, 1)$ has PDF:

$$\phi(z) = \frac{1}{\sqrt{2\pi}} e^{-z^2/2}$$

and CDF $\Phi(z) = P(Z \leq z)$.

Upper-Tail Probabilities: $P(Z > z)$

Commonly Used Critical Values:

B.1.2 Poisson Distribution

The Poisson distribution $X \sim \text{Pois}(\lambda)$ gives $P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$.

Poisson Probabilities for Selected Values of $\lambda$

Note: The values $\lambda = 1.2$ and $\lambda = 1.5$ are particularly relevant for soccer, as they approximate typical home and away goal-scoring rates in major European leagues (see Section B.3).

Cumulative Poisson Probabilities $P(X \leq k)$ for $\lambda = 1.5$ (typical home goals)

B.1.3 Binomial Distribution

The binomial distribution $X \sim \text{Bin}(n, p)$ gives $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$.

Binomial Probabilities for $n = 10$ (e.g., 10 shots on target)

Note: A shot-on-target conversion rate of $p = 0.30$ is a useful benchmark for top strikers in major leagues.

B.1.4 Student's $t$-Distribution

The $t$-distribution with $\nu$ degrees of freedom is used for inference about means with unknown population variance. As $\nu \to \infty$, $t_\nu \to \mathcal{N}(0,1)$.

Critical Values $t_{\alpha, \nu}$ (Upper-Tail)

For a two-tailed test at significance level $\alpha$, use the $\alpha/2$ column.

B.1.5 Chi-Squared Distribution

The chi-squared distribution $\chi^2_\nu$ with $\nu$ degrees of freedom. Used for goodness-of-fit tests, tests of independence, and comparing nested models.

Critical Values $\chi^2_{\alpha, \nu}$ (Upper-Tail)

B.1.6 F-Distribution Selected Critical Values

The $F$-distribution $F_{\nu_1, \nu_2}$ is used for ANOVA and comparing regression models.

$F_{0.05, \nu_1, \nu_2}$ Critical Values

B.2 Critical Values Quick Reference

B.2.1 Common Significance Thresholds

B.2.2 Sample Size Guidelines for Soccer Analytics

These rules of thumb help determine when certain approximations are reasonable:

B.3 Soccer Benchmarks

B.3.1 Goals Per Game by League

Average goals per match across Europe's top five leagues (approximate values; season-to-season variation of roughly $\pm 0.15$).

B.3.2 Expected Goals (xG) Benchmarks

B.3.3 Passing Benchmarks

Note: Passing statistics are heavily position-dependent. Center-backs and defensive midfielders typically have higher completion rates but fewer progressive passes than attacking players.

B.3.4 Defensive Benchmarks

B.3.5 Possession and Territorial Benchmarks

B.3.6 Physical Performance Benchmarks (per 90 minutes)

Physical benchmarks vary considerably by position. Full-backs and wide midfielders typically cover the most high-speed distance, while center-backs cover the least.

B.3.7 Goalkeeper Benchmarks

B.3.8 Common Scoreline Probabilities

Based on a Poisson model with $\lambda_{\text{home}} = 1.5$ and $\lambda_{\text{away}} = 1.15$ (assuming independence):

Note: The independence assumption (home goals independent of away goals) is a simplification. In practice, game state effects, tactical adjustments, and other factors introduce dependencies. Chapter 7 discusses more sophisticated approaches.

For interactive computation of these distributions, see the Python functions in Appendix C. For the mathematical foundations underlying these distributions, see Appendix A, Section A.2.