Statistical Analysis
F-Distribution Right-Tail P-Value & ANOVA Critical Cutoff (F.DIST.RT & F.INV.RT)
Convert ANOVA and variance-equality F-ratios into normalized Paulson cube-root Z-scores, right-tail p-values (F.DIST.RT), and alpha significance cutoffs across numerator and denominator degrees of freedom.
Translating raw F-ratios into right-tail p-values across degrees of freedom
In one-way ANOVA, regression overall-significance tests, and two-instrument precision audits, the ratio F = MS_between / MS_within follows Snedecor's F(df_1, df_2) distribution. Raw F-ratios cannot be judged against a fixed constant because small denominator degrees of freedom df_2 produce a heavy right tail.
Paulson's (1942) cube-root transformation Z = ((1 - 2/(9*df_2)) * F^(1/3) - (1 - 2/(9*df_1))) / SQRT((2/(9*df_2)) * F^(2/3) + 2/(9*df_1)) maps F(df_1, df_2) onto the standard normal scale accurately for df_2 >= 4, yielding right-tail p-values and alpha = 0.05 rejection flags.
| Observed F | df_1 (Num) | df_2 (Den) | Paulson Z | Right-Tail P-Value |
|---|---|---|---|---|
| 1.0000 | 8 | 18 | -0.0680 | 0.5271 (No effect) |
| 1.0000 | 18 | 18 | 0.0000 | 0.5000 (Exact median) |
| 8.0000 | 8 | 18 | 3.6111 | 0.0002 (Reject H0 at 1%) |
Paulson Cube-Root F-to-Z Transformation
With a_term = 2 / (9 * df_1) and b_term = 2 / (9 * df_2), evaluate F^(1/3) to standardize skewness across unequal degrees of freedom.
=LET(f_stat, B2, df_a, B3, df_b, B4, a_term, 2 / (9 * df_a), b_term, 2 / (9 * df_b), f_cbrt, f_stat ^ (1 / 3), ROUND(((1 - b_term) * f_cbrt - (1 - a_term)) / SQRT(b_term * f_cbrt ^ 2 + a_term), 4))Right-Tail Significance P-Value & Alpha Decision
Evaluate the upper-tail normal complement 1 - Phi(Z) and flag Reject_H0 whenever P_Right_Tail < Alpha.
=IF(E3 < B5, "Reject_H0", "Retain_H0")Try the example
Use the original values or change them to test your understanding.