1. Two objects, one name
A reader noted that ratio is among the oldest constructions in mathematics, and takes my article to argue that ratios are invalid or biased. My article makes no such claim. My issue focus on the bias of summary statistics (such as mean, standard deviation, or quartiles) if the intercept of the bivariate relationship between X and Y is nonzero.
The first object is the ratio itself, one number for each company. Under the statistical relation Y(i)=\alpha+\beta X(i), with X(i)>0 (the error term is dropped for algebraic simplicity):
\quad
(1)\qquad
m(i)=\frac{Y(i)}{X(i)}=\beta+\frac{\alpha}{X(i)}The second object about equation (1) is the univariate summary statistics: the single number obtained by pooling the N ratios into a mean, a standard deviation, a quartile, or an interquartile range, and then reading that number as an estimate of the economic parameter \beta. The article concerns the second object alone.
2. What “biased” means here
Bias is a property of an estimator relative to a stated target. It is not a property of a number. From (1), averaging over i:
\begin{aligned}
&(2)\quad
\mathbb{E}\!\left[\bar{m}\right]=\beta+\frac{\alpha}{X_{H}}
\\[8pt]
&\quad
X_{H}=\left[\frac{1}{N}\sum_{i=1}^{N}\frac{1}{X(i)}\right]^{-1}
\end{aligned}Two statements follow, both true. The mean of ratios is an unbiased estimator of \mathbb{E}\!\left[m\right], which is the quantity in (2). The mean of ratios is a biased estimator of \beta, by exactly \alpha/X_{H}. They differ only in the target named.
The transfer pricing is a question about \beta: the return an uncontrolled party earns on the denominator. Once that target is fixed, the summary statistic is biased for it. The defect lies in the pairing of statistic with target, not in the division that produced m(i).
3. The pooling step, not the division step
A univariate summary statistics presumes that the items summarized are draws from a common distribution. Write the relation with a disturbance, Y(i)=\alpha+\beta X(i)+\varepsilon(i), with \mathbb{E}\!\left[\varepsilon(i)\mid X(i)\right]=0 and \operatorname{Var}\!\left[\varepsilon(i)\mid X(i)\right]=\sigma^{2}. Dividing through by X(i):
\begin{aligned}
&(3)\quad
\mathbb{E}\!\left[m(i)\mid X(i)\right]=\beta+\frac{\alpha}{X(i)}
\\[8pt]
&\quad
\operatorname{Var}\!\left[m(i)\mid X(i)\right]=\frac{\sigma^{2}}{X(i)^{2}}
\end{aligned}Neither the conditional mean nor the conditional variance is the same for two companies of different size. The N ratios are not N observations on one quantity. They are N observations on N distinct quantities that share only \beta. Pooling them into a single location statistic is where the error enters. Division is innocent; aggregation is not.
4. Dispersion of ratios measures size, not profitability
Write u(i)=1/X(i), so that m(i)=\beta+\alpha\,u(i) is affine in u. Every dispersion statistic then passes through the affine map with \beta removed as a common additive constant:
\begin{aligned}
&(4)\quad
\operatorname{Var}(m)=\alpha^{2}\operatorname{Var}(u)
\\[8pt]
&\quad
\operatorname{sd}(m)=|\alpha|\operatorname{sd}(u)
\\[8pt]
&\quad
\mathrm{IQR}(m)=|\alpha|\,\mathrm{IQR}(u)
\end{aligned}The slope has cancelled identically in all three. Whatever the standard deviation or the interquartile range of a ratio indicator measures, it is not variation in profitability. Under (1) it is variation in the reciprocal of size, scaled by the intercept.
The eight-company illustration of the article makes the magnitude plain. There \alpha=5.0 and \beta=0.06, and the sample standard deviation of 1/X is 0.032862, so that:
\quad
(5)\qquad
\operatorname{sd}(m)=5.0\times 0.032862=0.1643A dispersion of 16.43 percentage points around a true margin parameter of 6.00 percent, in data containing no sampling error of any kind. The dispersion is not noise and it is not economics. It is \alpha multiplied by the spread of company size.
5. Antiquity does not authorize the summary
The classical theory of ratio, in Euclid V, Definition 5, makes sameness of ratio a statement about equimultiples: a ratio is invariant when both magnitudes are scaled together, so that A:B is the same ratio as \lambda A:\lambda B. Apply that test to the profit indicator. Scale the denominator by \lambda>0, so that Y becomes \alpha+\beta\lambda X(i):
\quad
(6)\qquad
m_{\lambda}(i)=\frac{\alpha+\beta\lambda X(i)}{\lambda X(i)}
=\beta+\frac{\alpha}{\lambda X(i)}and m_{\lambda}(i)=m(i) for all \lambda if and only if \alpha=0. When the intercept is nonzero the profit indicator is not scale invariant, and by the classical criterion it is not a ratio of magnitudes at all: it carries an absolute magnitude, \alpha, inside it. The reader who invokes the antiquity of ratio invokes Euclid’s equimultiple criterion, and the profit indicator satisfies that criterion when α = 0 and fails it otherwise. The classical and the statistical conditions are reached from two directions: the scale invariance that makes m(i) a ratio in Euclid’s sense is the same α = 0 that makes its mean and its quartiles estimate β.
6. The familiar case
Speed is distance divided by time, a ratio of unimpeachable pedigree. Consider two legs of equal distance d, driven at 30 and at 60 miles per hour. The arithmetic mean of the two ratios is:
\quad
(7)\qquad
\frac{30+60}{2}=45The speed of the trip is total distance over total time, and that quantity is the harmonic mean of the leg speeds:
\quad
(8)\qquad
\frac{2d}{\dfrac{d}{30}+\dfrac{d}{60}}
=\frac{2}{\dfrac{1}{30}+\dfrac{1}{60}}=H(30,60)=40The arithmetic mean of the leg speeds exceeds the harmonic mean by five miles per hour, and it is the harmonic mean that provides the reliable answer. The inversion is the one that governs (2): the ratio indicator averages 1/X(i) rather than X(i), and the reciprocal of that average is the harmonic mean X_{H}. Legs of equal distance here play the part that the intercept plays there, in that each fixes the weights under which the reciprocals are averaged.
No one reads the speed example as an objection to the concept of speed. The mechanism differs from the intercept problem of (2): here the discrepancy arises from the weighting implicit in the target rather than from a nonzero intercept. The structure is the same. The ratio is sound, the arithmetic mean of ratios is a well defined number, and that number answers a question that was not asked.
7. What is claimed, and what is not
Not claimed:
- that ratios cannot be computed, reported, or relied upon;
- that the division Y(i)/X(i) is invalid or meaningless;
- that ratio profit indicators are misleading in all circumstances.
Claimed:
- that the mean, standard deviation, and quartiles of m(i), read as estimates of \beta, are biased by \alpha/X_{H} and \alpha/X_{H,p} respectively, exactly and without asymptotic relief;
- that the dispersion statistics of m(i) are proportional to |\alpha| and to the spread of 1/X, and are independent of \beta;
- that the quartile ranking of m(i) is the inverse ranking of size when \alpha>0;
- that the condition under which these statistics are sound is \alpha=0, which is testable rather than assumed.
8. The test
Estimate the levels regression and inspect \hat{\alpha}\pm SE(\hat{\alpha}). Where the intercept is not distinguishable from zero, ratio statistics are defensible and I say so; where it is, they are biased in the exact amounts derived. Report the slope with its standard uncertainty, \hat{\beta}\pm SE(\hat{\beta}), on the pooled N=nT observations for n comparables over T years, with HC3 standard errors.
Generalized least squares (GLS) applied to the ratio equation, weighting each observation by X(i)^{2} as (3) requires, reduces algebraically to ordinary least squares (OLS) on the levels (Aitken, 1935). The regression is therefore not an alternative to the ratio analysis. It is the ratio analysis, carried out with the intercept estimated rather than set to zero by assumption.
References
Aitken, A. C. (1935). On least squares and linear combinations of observations. Proceedings of the Royal Society of Edinburgh, 55, 42–48. Canonical statement of generalized least squares (GLS); supplies the equivalence invoked in section 8.
Euclid. The Thirteen Books of the Elements, Book V, Definition 5. T. L. Heath (trans.), 2nd ed., Cambridge University Press, 1926. The equimultiple criterion for sameness of ratio; the invariance condition tested in equation (6).
Silva, E. (2026). The harmonic mean bias of ratio profit indicators. EdgarStat blog, September 8. The article to which this note responds.
Weisberg, S. (2014). Applied Linear Regression, 4th ed. Wiley, Hoboken. Standard reference for the levels regression and for weighted least squares under a known skedastic function.