Distribution Tables

Before software, statisticians read probability tables to turn a distribution into a number (a probability or a critical value). You will mostly use R — pnorm, qt, and friends — but reading a table by hand cements what those functions actually do, and many exams still hand you one. This appendix explains how to read each table and then generates small reference tables with live R code so the values are computed, never transcribed.

# Round helper so every printed table is tidy and reproducible.
fmt <- function(x, d = 4) formatC(x, format = "f", digits = d)

11. The standard Normal (z) table¶

The z-table gives the area to the left of a z-score under the standard Normal curve $N(0, 1)$. To find the area to the right, subtract from 1; the curve’s symmetry means the area below $-z$ equals the area above $+z$.

• R for an area (left tail): pnorm(z)

• R for a critical value: qnorm(area_to_left)

A two-sided $C\%$ confidence interval uses the critical value $z^\star$ that puts $(1-C)/2$ in each tail, i.e. qnorm(1 - (1 - C)/2).

z   <- c(-2, -1.5, -1, -0.5, 0, 0.5, 1, 1.5, 1.96, 2, 2.5)
data.frame(z = z, `area_left_P(Z<z)` = fmt(pnorm(z)), check.names = FALSE)

conf <- c(0.80, 0.90, 0.95, 0.98, 0.99)
data.frame(
  confidence_level = paste0(conf * 100, "%"),
  tail_area_each   = fmt((1 - conf) / 2),
  z_star           = fmt(qnorm(1 - (1 - conf) / 2))
)

The 68–95–99.7 rule is just three z-areas; here are the exact figures:

k <- 1:3
data.frame(
  within_k_SD = k,
  exact_area  = fmt(pnorm(k) - pnorm(-k)),
  rounded     = c("68%", "95%", "99.7%")
)

22. The t table¶

The t-distribution is used for inference about means when $\sigma$ is unknown. It is bell-shaped but heavier-tailed than the Normal; its shape depends on the degrees of freedom ($df$). As $df$ grows, the t-table converges to the z-table (look at the bottom row below).

• R for a critical value: qt(1 - tail_area, df)

• A two-sided $C\%$ CI for a mean uses qt(1 - (1 - C)/2, df) with $df = n-1$.

dfs   <- c(1, 5, 10, 15, 20, 30, 50, 100, Inf)
levels_c <- c(0.90, 0.95, 0.99)
tbl <- sapply(levels_c, function(C) qt(1 - (1 - C) / 2, dfs))
out <- data.frame(df = ifelse(is.finite(dfs), as.character(dfs), "Inf (= z)"),
                  check.names = FALSE)
out[paste0("t*_", levels_c * 100, "%")] <- fmt(tbl)
out

33. The chi-square (χ²) table¶

The chi-square distribution is right-skewed and lives on $[0, \infty)$. We use it for goodness-of-fit and independence tests, which are always one-sided (upper tail): only large $\chi^2$ values count as evidence against the null.

• R for an upper-tail critical value: qchisq(1 - alpha, df)

• R for a p-value from a statistic: pchisq(stat, df, lower.tail = FALSE)

Degrees of freedom: $k - 1$ for goodness of fit ($k$ categories); $(r-1)(c-1)$ for an $r \times c$ independence table.

df_chi <- c(1, 2, 3, 4, 5, 6, 8, 10)
alphas <- c(0.10, 0.05, 0.01)
m <- sapply(alphas, function(a) qchisq(1 - a, df_chi))
out <- data.frame(df = df_chi)
out[paste0("alpha=", alphas)] <- fmt(m, 3)
out

44. The F table¶

The F-distribution is used in ANOVA to compare variances. It is right-skewed and indexed by two degrees of freedom: the numerator $df_1 = k - 1$ (number of groups minus one) and the denominator $df_2 = N - k$ (total observations minus number of groups). ANOVA tests are upper-tailed: large $F$ means the group means differ.

• R for a critical value: qf(1 - alpha, df1, df2)

• R for a p-value: pf(F_stat, df1, df2, lower.tail = FALSE)

Because $F$ needs two $df$’s, a printed table fixes $\alpha$ (here 0.05) and lays $df_1$ across the top, $df_2$ down the side:

df1_vals <- c(1, 2, 3, 4, 5)
df2_vals <- c(5, 10, 15, 20, 30, 60, 120)
fmat <- outer(df2_vals, df1_vals, function(d2, d1) qf(0.95, d1, d2))
out <- data.frame(df2 = df2_vals)
out[paste0("df1=", df1_vals)] <- fmt(fmat, 3)
out

55. Quick R reference for every table¶