Appendix B: Statistical Tables

This appendix provides reference tables for the most commonly used statistical distributions in sports betting analysis. While modern software makes table lookups largely unnecessary for computation, these tables remain valuable for building intuition, checking quick calculations, and understanding critical values referenced throughout the text.

B.1 Standard Normal (Z) Distribution Table

The table gives P(Z <= z) for the standard normal distribution N(0,1). For negative z values, use the symmetry property: P(Z <= -z) = 1 - P(Z <= z).

Commonly used critical values:

Betting application: To test whether a bettor's 55% win rate over 400 bets at -110 odds (break-even = 52.38%) is statistically significant: z = (0.55 - 0.5238) / sqrt(0.5238 * 0.4762 / 400) = 0.0262 / 0.02496 = 1.05. Since 1.05 < 1.96, we cannot reject the null at the 5% level. The result is not statistically significant.

B.2 Student's t-Distribution Critical Values

The table gives t_{alpha, nu} such that P(T > t) = alpha for a t-distribution with nu degrees of freedom. For two-tailed tests, use the column for alpha/2.

Betting application: When comparing the mean ROI of two betting strategies over a small sample of 15 weeks, use df = 14 for the paired t-test. At alpha = 0.05 (two-tailed), the critical value is t_{0.025, 14} = 2.145 (interpolated between df=12 and df=15).

B.3 Chi-Squared Distribution Critical Values

The table gives chi^2_{alpha, k} such that P(X > chi^2) = alpha for the chi-squared distribution with k degrees of freedom.

Betting application: A calibration test (Chapter 17) bins predicted probabilities into 10 groups and compares observed frequencies to expected frequencies using a chi-squared goodness-of-fit test with df = 8 (10 bins minus 2 for estimated parameters). At alpha = 0.05, the critical value is 15.507. A test statistic exceeding this suggests the model is poorly calibrated.

B.4 F-Distribution Critical Values (Selected)

The table gives F_{alpha, d1, d2} for alpha = 0.05 (upper tail). d1 = numerator degrees of freedom (columns); d2 = denominator degrees of freedom (rows).

Betting application: The F-test is used in ANOVA to compare the predictive power of multiple models simultaneously (Chapter 16). When comparing 4 models on 60 games, use d1 = 3 (4-1) and d2 = 56 (60-4). The critical value at alpha = 0.05 is approximately 2.77.

B.5 Common Probability Distribution Properties

B.6 Sample Size Requirements for Betting Analysis

The following table provides the minimum number of bets required to detect a given edge at the 95% confidence level (alpha = 0.05, two-tailed) with 80% power (beta = 0.20), assuming standard -110 odds (break-even at 52.38%).

Formula used: n = (z_{alpha/2} + z_{beta})^2 * p_0*(1-p_0) / (p_1 - p_0)^2, where p_0 = 0.5238 and p_1 is the true win rate.

Key insight for bettors: Even a genuinely skilled bettor with a 55% win rate at -110 needs over 2,200 bets to achieve statistical significance. This explains why many profitable bettors cannot "prove" their edge to skeptics within a single season. The variance inherent in betting outcomes makes short-term results nearly meaningless for distinguishing skill from luck.

Additional Sample Size Guidelines

B.7 Quick Reference: Percentiles of the Standard Normal

For rapid mental math in betting contexts:

Point spread conversion: If point spreads are modeled as Normal(spread, sigma) with sigma approximately 13.5 points (NFL), a 3-point favorite has P(cover) = P(Z > -3/13.5) = P(Z > -0.222) = 0.588, or approximately 59% against the spread before accounting for the vig.

For computations requiring greater precision than these tables provide, use the scipy.stats module in Python as described in Appendix C.