Appendix B: Statistical Tables

These tables give the classic critical values for the three distributions this book uses over and over: the t distribution (inference for a single slope or mean), the F distribution (the ANOVA table, the general linear test, comparing full and reduced models), and the chi-square distribution (goodness-of-fit style tests, some variance-comparison and categorical-data procedures, and the reference distribution behind the deviance in Chapters 13 and 14). Every number below was computed directly by calling scipy.stats in Python, not copied from an older printed table. Because table lookup only ever finds you approximate df values, R and Python compute the exact critical value for any df you actually have. The one line calls that replace each table appear under each section, so you never need to interpolate between rows again.

When you need each table¶

• t table. You need a critical value t* when you build a confidence interval or run a hypothesis test for one coefficient, using b_1 or another single estimate and its standard error. This is the table behind every “t inference” section in Chapters 2 to 3, 8, and beyond.

• F table. You need a critical value F* whenever you compare two nested models: the overall regression F test in the ANOVA table (3.7 The F test and its equivalence to the t test), or the general linear test comparing a full model against a reduced one (8.4 The general linear test). The F table is really two numbers, the numerator df and the denominator df, so it is printed as a grid.

• Chi-square table. You need a critical value chi-square* for tests whose reference distribution is chi-square rather than t or F: a variance ratio test, a goodness-of-fit test, or a likelihood ratio test in the logistic and Poisson chapters (Chapters 13 and 14), where twice the log-likelihood difference between two nested models is compared to a chi-square distribution on the difference in the number of parameters.

Software has replaced the printed table¶

Every value in these tables can be reproduced, for any df or any tail area, not just the ones printed here, with one line of code. In words: instead of rounding your degrees of freedom to the nearest tabled row, you ask the software for the exact critical value at your exact df.

qt(0.975, df = 23)      # t critical value, two-sided alpha = 0.05, df = 23
qf(0.95, df1 = 3, df2 = 20)   # F critical value, alpha = 0.05, 3 and 20 df
qchisq(0.95, df = 5)    # chi-square critical value, alpha = 0.05, df = 5

from scipy import stats
stats.t.ppf(0.975, df=23)       # t critical value, two-sided alpha = 0.05, df = 23
stats.f.ppf(0.95, dfn=3, dfd=20)     # F critical value, alpha = 0.05, 3 and 20 df
stats.chi2.ppf(0.95, df=5)      # chi-square critical value, alpha = 0.05, df = 5

These printed tables remain useful for a quick sanity check by hand, for exams where you work without a computer, and for seeing at a glance how a critical value shrinks as df grows. Use the software call whenever you have an exact df in front of you.

t table¶

Two-sided critical values t* such that P(|T| > t*) equals the stated alpha, for a t random variable on the given degrees of freedom. In words: this is the multiplier you attach to a standard error to build a two-sided confidence interval, or the cutoff you compare |t| against in a two-sided test. The one-sided value at alpha = 0.05 is the column headed t(.05, one-sided); for a one-sided test at alpha = 0.01, use the t(.01, one-sided) column. The last row (df = inf) is the standard normal z, the limit as df grows without bound.

Software. qt(1 - alpha, df) in R, stats.t.ppf(1 - alpha, df) in Python, where alpha is the one-tailed area (0.025 for a two-sided 95 percent interval, 0.05 for the first column above).

F table, alpha = 0.05¶

Upper 5 percent critical values F*, indexed by numerator degrees of freedom df1 (across the top, the model or extra-parameter df) and denominator degrees of freedom df2 (down the side, the error df). In words: if your computed F statistic exceeds the value in the cell matching your df1 and df2, you reject at alpha = 0.05.

F table, alpha = 0.01¶

Same layout, upper 1 percent critical values.

Software. qf(1 - alpha, df1, df2) in R, stats.f.ppf(1 - alpha, dfn=df1, dfd=df2) in Python, for any pair of df you have, including values not printed above.

Chi-square table¶

Right-tail critical values chi-square* such that P(X > chi-square*) equals the column heading, for a chi-square random variable on the given df. In words: this is the cutoff you compare a chi-square statistic or a likelihood ratio statistic against; the small-area columns on the left (0.995, 0.99) matter mainly for two-sided variance tests, and the columns on the right (0.05, 0.025, 0.01, 0.005) are the ones you use most often for a one-sided goodness-of-fit or deviance test.

Software. qchisq(1 - alpha, df) in R, stats.chi2.ppf(1 - alpha, df) in Python, for any df, including values well beyond the last row printed here.

How these tables were produced¶

Every value above came from a script that calls scipy.stats.t.ppf, scipy.stats.f.ppf, and scipy.stats.chi2.ppf and formats the result as Markdown; the script is book-myst/appendix/_gen_tables.py. No value was typed by hand or copied from another book.