Back-Calculation of Fish Lengths Based on Proportionality between Scale and Length Increments

Abstract

The assumption underlying linear back-calculation offish lengths (L) from scale measurements (S) is that annual increments of length (ΔL) are proportional to annual increments between the annuli on scales (ΔS) for each fish–scale combination. A sample from a fish population that provides measurements of fish length and scale radius at time of collection should be symmetrical transversely to its central axis when plotted with an absolute slope of 45°; if not, it can be made symmetrical by either of two easy methods. The slope of the central axis can be estimated using either an arithmetic mean regression or (preferably) the geometric mean regression; either should be made to pass through the centroid of the whole sample. Average lengths at each age are unbiased if computed from average annulus distances using one of the above symmetrical statistics whereas biased estimates are obtained using either of the ordinary regressions between S and L, especially when part of the range of ages (older than 0) is missing at either end of the sample. Lengths of individual fish can be back-calculated throughout their life by using either the Whitney–Carlander or (preferably) the Fraser–Lee procedure, with the fixed parameter estimated from a symmetrical regression line.