Appendix B: Statistical Tables

This appendix provides essential statistical reference tables for hypothesis testing, confidence interval construction, and probability calculations commonly used in basketball analytics.

B.1 Standard Normal Distribution (Z-Table)

The standard normal distribution table provides the cumulative probability P(Z <= z) for the standard normal random variable Z with mean 0 and standard deviation 1. Use this table for large-sample hypothesis tests and confidence intervals.

How to Read This Table

The z-value is split into two parts: the row indicates the ones and tenths digits, while the column indicates the hundredths digit. For example, to find P(Z <= 1.96), locate row 1.9 and column 0.06 to get 0.9750.

Cumulative Probabilities P(Z <= z)

Negative Z-Values

For negative z-values, use the symmetry property: P(Z <= -z) = 1 - P(Z <= z)

For example: P(Z <= -1.96) = 1 - 0.9750 = 0.0250

Common Critical Values for Two-Tailed Tests

Basketball Application Example: Testing if a player's three-point percentage (38.5% on 200 attempts) differs significantly from the league average (35.0%). Use z = (0.385 - 0.350) / sqrt(0.35 * 0.65 / 200) = 1.04, which is less than 1.96, so we do not reject the null hypothesis at alpha = 0.05.

B.2 Student's t-Distribution Critical Values

The t-distribution is used when the population standard deviation is unknown and must be estimated from the sample. This table provides critical values for various degrees of freedom (df) and significance levels.

Two-Tailed Critical Values

One-Tailed Critical Values

For one-tailed tests, use the column header divided by 2. For example, for a one-tailed test at alpha = 0.05, use the t(0.10) column.

Basketball Application Example: Testing if the mean points per game for a sample of 25 games (mean = 112.4, s = 8.5) differs from a hypothesized mean of 108.0. With df = 24, t = (112.4 - 108.0) / (8.5 / sqrt(25)) = 2.59. Since 2.59 > 2.064 (t-critical at alpha = 0.05), we reject the null hypothesis.

B.3 Chi-Square Distribution Critical Values

The chi-square distribution is used for goodness-of-fit tests, tests of independence, and variance tests. This table provides critical values for the right-tail probability.

Right-Tail Critical Values

Basketball Application Example: Testing if shot distribution across 5 court zones follows the expected distribution. With df = 4, if the calculated chi-square statistic is 12.5 and the critical value at alpha = 0.05 is 9.488, we reject the null hypothesis of uniform distribution.

B.4 F-Distribution Critical Values

The F-distribution is used in ANOVA and regression analysis. This table provides critical values at alpha = 0.05 for the most common degrees of freedom combinations.

F-Critical Values at alpha = 0.05

Basketball Application Example: Testing if the mean points scored differ significantly across 4 quarters using one-way ANOVA with 20 games. With df1 = 3 and df2 = 76, the critical F-value at alpha = 0.05 is approximately 2.72.

B.5 Correlation Critical Values

Critical values for the Pearson correlation coefficient to test H0: rho = 0.

Two-Tailed Critical Values

Basketball Application Example: With a sample of n = 30 games, testing the correlation between assists and team wins. If r = 0.42, the critical value at alpha = 0.05 is 0.361. Since 0.42 > 0.361, the correlation is statistically significant.

B.6 Binomial Probabilities

Selected binomial probabilities P(X = k) for common n and p values relevant to basketball analytics.

n = 10 (e.g., 10 free throw attempts)

n = 20 (e.g., 20 field goal attempts)

Basketball Application Example: A player with 80% free throw percentage attempts 10 free throws. The probability of making exactly 8 is 0.3020, making at least 8 is 0.3020 + 0.2684 + 0.1074 = 0.6778.

B.7 Poisson Distribution Probabilities

The Poisson distribution models count data for rare events. Useful for modeling turnovers, steals, and blocks per game.

P(X = k) for various lambda values

Basketball Application Example: If a team averages 3 blocks per game (lambda = 3), the probability of getting exactly 5 blocks is 0.1008, and the probability of getting 5 or more blocks is approximately 0.1847.

B.8 Quick Reference: Common Hypothesis Testing Scenarios

One-Sample Tests

Two-Sample Tests

ANOVA

B.9 Sample Size Determination

Sample Size for Estimating a Mean

$$n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2$$

where E is the margin of error.

Sample Size for Estimating a Proportion

$$n = \frac{z_{\alpha/2}^2 \cdot p(1-p)}{E^2}$$

For maximum variance (p = 0.5):

Basketball Application Example: To estimate a player's true three-point percentage within 3 percentage points with 95% confidence, you need approximately n = (1.96)^2 * (0.35)(0.65) / (0.03)^2 = 971 attempts.

B.10 Effect Size Reference Tables

Cohen's d Interpretation

Correlation Coefficient Interpretation

R-squared Interpretation for Regression

Note: In basketball analytics, R-squared values tend to be lower due to high game-to-game variance. An R-squared of 0.15-0.25 for single-game predictions may be considered reasonable.

B.11 Power Analysis Reference

Statistical power is the probability of correctly rejecting a false null hypothesis. This table shows required sample sizes for various power levels when detecting a medium effect size (d = 0.5) with alpha = 0.05.

Sample Size per Group for Two-Sample t-Test

Basketball Application Example: To detect a difference of 2 points per game (approximately d = 0.5 with typical game variance) with 80% power, you need approximately 64 games per group being compared.

This appendix provides reference values for common statistical procedures. For calculations beyond these tables, use statistical software such as Python's scipy.stats module or R's built-in distribution functions.