Median: The Mid-Point of the Comparables Distribution

Definition

The median is the middle value of the comparables’ PLI distribution — the value at which half the pool’s members sit below and half sit above (the 50th percentile). In the arm’s length range it is the mid-point — the centre of the IQR (the 25th and the 75th percentiles, the median between them) — and the tested party’s position against it is the standard’s first read: inside the IQR, the distance from the mid-point, the direction (above, below).

The pool's PLI values (n members, sorted):
  … p25 … **median (p50)** … p75 …
          the mid-point of the IQR
          the tested party’s position is measured against it

The use	The content
The IQR’s centre	The median is the IQR’s mid-point — the range is p25–p75, the mid-point is the median, the tested party’s position stated as the IQR relation and the mid-point distance
The full-range practice	Where the full range (the min–max, the p5–p95 on the stated convention) is the range, the median is still the distribution’s centre — the IQR vs full range guide has the convention’s choice
The sensitivity read	The median’s stability against the outliers (the median is robust to the tails, the mean is not) — the distribution’s shape (the skew, the gap) read with the median as the centre

The working read (the IQR vs full range guide): the tested party’s position is documented as the IQR relation (inside / outside, the bound) and the mid-point distance (the percentage points from the median, the direction) — the two reads together. The mid-point is not a separate standard (the IQR is the arm’s length range, the OECD’s convention) — it is the position measure inside the range, and the distance from it is the number the TPO and the appeal argue on (the tested party at the IQR edge, 2 points from the mid-point, is a different file than the tested party at the mid-point).

Example

The pool (12 comparables, the adjusted PLIs): the p25 is 2.1%, the median is 2.8%, the p75 is 3.4% — the IQR is 2.1%–3.4%, the mid-point 2.8%. The tested party’s adjusted PLI is 3.1%: inside the IQR, 0.3 points above the mid-point — the position documented (the IQR relation: inside; the mid-point distance: +0.3). The distribution’s shape (the 12 values, the no-material-outlier read) supports the IQR’s validity — the statistical significance of the set noted, the dispersion (the spread, the p75−p25 width relative to the mid-point) read as the pool’s consistency.

See also

• Interquartile Range (IQR) · Percentile
• IQR vs Full Range
• Outlier · Statistical Significance

FAQ

Is the mid-point (the median) a separate arm’s length standard? No — the IQR is the arm’s length range (the OECD’s convention, the TP guidelines position); the mid-point is the range’s centre, and the tested party’s position against it is the position measure inside the range. The documentation states both — the IQR relation and the mid-point distance — because the appeal argues on the distance as much as the relation.

Why the median and not the mean as the centre? The median is robust to the tails — the outlier at the p98 moves the mean, not the median — and the comparables distribution is exactly the context where the tails are the question (the one high-margin member, the one loss-making member). The IQR itself is the robust range (the p25–p75, the middle 50%); the median is its consistent centre. The mean has no role in the standard — it is the number the distribution’s shape describes, not the number the standard measures.

Does the multi-year data change the median? The median is computed on the pool’s PLI values as the method produces them — the single-year values, or the multi-year averaged values (the per-member average, on the stated averaging method), depending on the study’s method. The median is the distribution’s centre on the values the method carries — the averaging is the method’s choice, stated; the median is the statistic on the result.