Appendix A: Statistical Tables

A note on tables in the age of Python and Excel: The tables in this appendix are included for three reasons: (1) they help you build intuition for how distributions behave, (2) they allow hand calculations during exams and practice, and (3) they provide quick reference values without opening a computer. In professional practice, you will almost always use scipy.stats in Python or built-in functions in Excel rather than looking up values in printed tables. That said, understanding what these tables contain makes you a better user of the software.

A.1 Standard Normal (z) Distribution Table

The table below gives the cumulative probability P(Z <= z) for the standard normal distribution, where Z ~ N(0, 1). To find the area to the LEFT of a given z-score, locate the row for the ones and tenths digits and the column for the hundredths digit.

How to use this table: - For P(Z <= z): Read the value directly. - For P(Z > z): Compute 1 - P(Z <= z). - For P(-z < Z < z): Compute 2 * P(Z <= z) - 1. - For P(Z < -z): By symmetry, P(Z < -z) = P(Z > z) = 1 - P(Z <= z).

Abbreviated Standard Normal Table (Negative z)

Abbreviated Standard Normal Table (Positive z)

Common Critical Values (Quick Reference)

Python equivalent:

from scipy.stats import norm
# P(Z <= z)
norm.cdf(1.96)           # 0.9750
# z* for a given confidence level
norm.ppf(0.975)          # 1.960 (two-tailed 95%)

A.2 Student's t-Distribution Critical Values

The table below gives the critical value t such that P(T > t) = alpha for a t-distribution with the specified degrees of freedom (df). For two-tailed tests, use the column header that represents alpha/2 in each tail.

How to use this table: - One-tailed test: Use the one-tail alpha column directly. - Two-tailed test: Use the two-tail alpha row. For example, for a 95% confidence interval, use the 0.025 column (alpha/2 = 0.025 in each tail). - Confidence intervals: The two-tailed alpha = 1 - confidence level. For 95% CI, alpha = 0.05, so use the 0.025 column.

Key observation: As df increases, t approaches z. For df > 30, the t-distribution is nearly indistinguishable from the standard normal. The last row (df = infinity) matches the z-table critical values exactly.

Python equivalent:

from scipy.stats import t
# Critical value for 95% CI with df = 20
t.ppf(0.975, df=20)      # 2.086
# p-value for t = 2.53 with df = 48 (two-tailed)
2 * t.sf(2.53, df=48)    # 0.0147

A.3 Chi-Square Distribution Critical Values

The table below gives the critical value chi-sq such that P(X^2 > chi-sq) = alpha for a chi-square distribution with the specified degrees of freedom. Reject H0 when the computed chi-square statistic exceeds the table value.

Degrees of freedom for chi-square tests: - Goodness-of-fit: df = k - 1 (where k = number of categories) - Test of independence: df = (r - 1)(c - 1) (where r = rows, c = columns)

Python equivalent:

from scipy.stats import chi2
# Critical value at alpha = 0.05 with df = 4
chi2.ppf(0.95, df=4)        # 9.488
# p-value for chi-sq = 12.4 with df = 4
chi2.sf(12.4, df=4)          # 0.0145

A.4 F-Distribution Critical Values (alpha = 0.05)

The F-distribution has two degrees of freedom parameters: df1 (numerator, between-groups) and df2 (denominator, within-groups). The table gives the critical value F such that P(F > F) = 0.05. Reject H0 in one-way ANOVA when the computed F-statistic exceeds the table value.

F Critical Values at alpha = 0.05

F Critical Values at alpha = 0.01

For one-way ANOVA: - df1 = k - 1 (number of groups minus 1) - df2 = N - k (total observations minus number of groups)

Python equivalent:

from scipy.stats import f
# Critical value at alpha = 0.05 with df1 = 3, df2 = 20
f.ppf(0.95, dfn=3, dfd=20)   # 3.098
# p-value for F = 4.75 with df1 = 3, df2 = 36
f.sf(4.75, dfn=3, dfd=36)     # 0.0069

A.5 Quick Reference: Which Table Do I Need?

When in doubt, use Python:

from scipy import stats
# Any distribution: stats.{dist}.cdf(), .ppf(), .sf()
# cdf(x) = P(X <= x)
# ppf(q) = x such that P(X <= x) = q  (inverse CDF)
# sf(x)  = P(X > x) = 1 - cdf(x)     (survival function)