1. The IQR does not shrink with sample size (n)
For a normal population (Abramowitz and Stegun 1964, ch. 26; Hoaglin, Mosteller and Tukey 1983), Q_1=\mu-0.6745\sigma and Q_3=\mu+0.6745\sigma. Therefore:
\mathrm{IQR}=Q_3-Q_1\approx 1.349\sigmaThe half-width 0.6745\sigma is the classical “probable error” (der wahrscheinliche Fehler), which the astronomer Friedrich Wilhelm Bessel introduced in 1815 and defined in 1816 (Hald 1998, p. 360). The IQR of a normal population is twice the probable error: a dispersion measure from nineteenth-century astronomy, not an estimator of a parameter.
As n grows, the sample IQR approaches 1.349\sigma. By contrast, the width of a one-standard-error interval for the sample mean is 2\sigma/\sqrt{n} (approximately 68% coverage under normality when \sigma is known).
\frac{\mathrm{IQR}}{2\sigma/\sqrt{n}}\approx 0.6745\sqrt{n},\quad \text{or}\quad \frac{\mathrm{IQR}}{\text{one-SE width}}\approx 0.6745\sqrt{n}The IQR measures cross-company variation; the one-SE interval measures estimation uncertainty. They answer different questions. More independent observations reduce uncertainty; the IQR converges to 1.349\sigma. It’s not more reliable than the standard error.
2. The sample median is less efficient under normality
\operatorname{Var}(\text{median})\approx(\pi/2)\sigma^2/n=1.571\sigma^2/n, versus \operatorname{Var}(\text{mean})=\sigma^2/n (Kendall and Stuart 1977). Under normality, the median needs about 57% more independent observations to achieve the same variance.
3. Ratio PLIs carry the intercept
If Y=a+bX, with X>0, then m=Y/X=b+a/X. For a sample of n companies:
\operatorname{mean}(m)=b+\frac{a}{H}where H=n/\sum_{i}(1/X_i) is the harmonic mean of X. When a\neq 0, every ratio differs from the slope (b) by the bias a/X_i.
The quartiles inherit the same bias. Because m=b+a/X is monotone in X, the rank order of m reverses that of X when a>0:
Q_p(m)=\begin{cases}b+\dfrac{a}{Q_{1-p}(X)} &\text{if }a>0\\ b+\dfrac{a}{Q_p(X)} &\text{if }a<0\end{cases}\mathrm{IQR}(m)=|a|\left[\frac{1}{Q_{25}(X)}-\frac{1}{Q_{75}(X)}\right]The location of the range shifts with the intercept (a), and its width is driven by the intercept and the spread of X, not by any spread in b. An IQR of ratios does not test whether a=0.
The regulatory alternative
Treas. Reg. §1.482-1(e)(2)(iii)(B) says that the IQR ordinarily provides an acceptable measure, but permits a different statistical method if it is more reliable. Regression on company-year observations can estimate the slope and test the intercept, while reporting an appropriate standard error for the coefficients.
Why one standard error?
A coefficient estimate accompanied by one standard error makes sampling precision transparent. Under a normal approximation, coefficient \pm 1 SE covers the true coefficient with probability of approximately 68.27%; for a slope regression with intercept on n = 30 comparables (n-2=28 degrees of freedom), the two-sided 68.27% t-multiplier is about 1.018. By comparison, the two-sided 95% multiplier is 2.306 at 8 df (n = 10) and 2.048 at 28 df (n = 30).
Like García Márquez’s Crónica de una muerte anunciada, boilerplate computations of PLI quartiles are headed to audit, appeals, or litigation defeat.
References
Abramowitz, M. and I. Stegun (eds.) (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Applied Mathematics Series 55. Washington, DC: U.S. Government Printing Office, ch. 26.
García Márquez, G. (1981). Crónica de una muerte anunciada. Bogotá: La Oveja Negra.
Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930. New York: Wiley.
Hoaglin, D., F. Mosteller and J. Tukey (eds.) (1983). Understanding Robust and Exploratory Data Analysis. New York: Wiley.
Kendall, M. and A. Stuart (1977). The Advanced Theory of Statistics, Vol. 1: Distribution Theory, 4th ed. London: Griffin.
Tukey, J. (1977). Exploratory Data Analysis. Reading, MA: Addison-Wesley.
U.S. Treasury Regulations, 26 C.F.R. §1.482-1(e)(2)(iii)(B).