RoyaltyStat Blog

Arm's Length Range - Most Reliable Measure

Posted by Ednaldo Silva on Mar 20, 2016 7:08:06 AM

In statistics, we consider a random variable in terms of a reliable estimate of its central value (center) and spread from the center. The center is measured by the mean, and the spread by the standard deviation from the mean. In practice, data samples (such as the usual judgement selection of comparables in transfer pricing practice) contain outliers that distort the estimates of mean and standard deviation. As a result, statistical (confidence or tolerance) intervals based on the mean and standard deviation may not be reliable. See Gerald Hahn & William Meeker, Statistical Intervals (John Wiley, 1991) for a detailed coverage of tolerance and confidence intervals.

In general, statistical intervals are based on a formula combining the best (i.e., most reliable) estimate of center together with a probable range around the center:

  • Best estimate of center = average ± Uncertainty
  • Uncertainty = Probable range (δ) measures data spread from center

where δ measures the displacement from the center. See John Taylor, Introduction to Error Analysis (2nd edition) (University Science, 1982), pp. 9, 13-14.

We can use two alternative strategies to control the measurable adverse effects of outliers. First, we can use robust measures of center and spread, such as using the median, trimean, and quartiles. Second, we can set aside the outliers before computing the mean and standard deviation. These strategies are analyzed in the treatise edited by David Hoaglin, Frederick Mosteller, and John Tukey, Understanding Robust and Exploratory Data Analysis (John Wiley, 1983).

U.S. Treas. Reg. § 1.482-1(e)(2)(iii)(B)(Arm’s length range) reflect these principles of using statistical intervals based on the most reliable measure of center and spread by providing for “adjusting the range through application of a valid statistical method to the results of all of the uncontrolled comparables selected.” Plus, mas: “The interquartile range ordinarily provides an acceptable measure of this range; however a different statistical method may be applied if it provides a more reliable measure.” Emphases added.

The interquartile range (IQR) is not likely to provide the most reliable measure of the arm’s length range because it produces wide Q1 (lower quartile) and Q3  (upper quartile) limits of the price or profit indicator used to determine arm's length transactions between related-parties. Therefore, in controversy, the U.S. regulatory misguidance that the IQR "ordinarily provides an acceptable measure" of the arm's length is likely to be subverted by the higher statistical (scientific) principle that the tax administration and taxpayer are obligated to provide the "most reliable" measure of the arm's length range.  Higher principle because the selection of the most reliable measure of center and spread has no exception in science or in the U.S. transfer pricing regulations. E.g., as a statistical or factual matter, use of the IQR can be subverted by using “notches” with the whole data sample (full selection of comparables) or confidence intervals based on trimmed (outlier free) data. Notches are discussed in David Hoaglin, et. al, op. cit., p. 76.

In RoyaltyStat, we incorporated several user-friendly tools to filter outliers and produce statistical intervals, including notches and confidence intervals for the mean and the median. Thoughtless use of the IQR is bound to be challenged in transfer pricing controversy. 

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Ednaldo Silva (Ph.D.) is founder and managing director at RoyaltyStat. He helped draft the US transfer pricing regulations and developed the comparable profits method called TNNM by the OECD. He can be contacted at:

RoyaltyStat provides premier online databases of royalty rates extracted from unredacted agreements, normalized company financials (income statement, balance sheet, cash flow), and annual reports. We provide high-quality data, built-in analytical tools, customer training and attentive technical support.

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Topics: Arm's Length Range