1. Biodiversity of Fishes Length-Weight Relationships
Rainer Froese
( )

2. How to Measure Size in Fishes
Length as a proxy for weight
Total length (TL; tip of snout to caudal fin end)
Standard length (SL; tip of snout to caudal peduncle)
Fork length (FL; tip of snout to mid of forked caudal fin)
Other length measurements (e.g. width in rays)
Length as proxy for size overestimates weight in eels, underestimates in puffers and boxfishes

3. Relationship Between Weight and Length
W = a * Lb
with weight W in grams and length L in cm, to standardize parameters
For parameter estimation use linear regression of data transformed to base 10 logarithms
log10 W = log10 a + b * log10 L
Plot data to detect and exclude outliers, and to check for growth stanzas

4. LWR Plot I
Weight-length data for Cod taken in 1903 by steam trawlers from Moray Firth and Aberdeen Bay.
Data were lumped by 0.5 cm length class and thus one point may represent here 1-12 specimens.

5. LWR Plot II log W = -2.206 + 3.11 * log L W = 0.0062 L3.11
a = –
b = 3.10 – 3.11
n = 468, r2 = 0.999
Double-logarithmic plot of the data in LWR Plot I. The overall regression line is W = * L3.11, with
n = 468, r2 = 0.999, 95% CL of a = – , 95% CL of b = – Note that the mid-length of length classes was used such as cm for the length class cm and the number of specimens per length class (1 - 12) was used a frequency variable in the linear regression.

6. Growth Stanzas
Double-logarithmic plot of weight vs. length for Clupea harengus, based on data in Fulton (1904), showing two growth stanzas and an inflection point at about 8 cm. For the first growth stanza: n = 5 (92), r2 = 0.998, 95% CL of a = – , 95% CL of b = 3.66 – For the second growth stanza: n = 46(400),
r2 = 0.999, 95% CL of a = – , 95% CL of b = 3.28 – 3.29.

7. How to Report LWR r2 = 0.994 W = 0.0121 * L3.03 n = 54 sex = mixed
Length range = cm TL
95% CL a = –
95% CL b = 2.99 – 3.09
Species: Gadus morhua Linnaeus, 1758
Locality: Kiel Bight, Germany
Gear: Bottom trawl with 6 cm mesh size.
Sampling duration: Mid April to mid May, 2005
Remarks: Beginning of spawning season.

8. Fulton’s Condition Factor
C = 100 * W / L3
Used to compare ‘fatness’ or condition of specimens (of the same species) of similar size, e.g. to detect differences between sexes, seasons or localities.
Example: Condition of a specimen of 10 grams weight and 10 cm length
1 = 100 * 10 / 1000

9. Relationship between a and C
If b is reasonably close to 3 (2.95 – 3.05) then C ≈ 100 * a and C is an average across the whole length range from which LWR was derived

10. Condition as a Function of Size and Season
b = 2.96
b = 2.91
b = 2.77
b = 2.60
Log-log plot of condition vs. length of Comber Serranus cabrilla taken in spring, summer, fall and
winter, respectively, in the Aegean Sea. The dotted line shows the condition factors associated
with geometric mean a and mean b across all available LWRs for this species (Froese 2006).
Spawning season is in spring.

11. Understanding b
Frequency distribution of mean exponent b based on 3,929 records for 1,773 species, with
median = 3.025, 95% CL = – 3.036, 5th percentile = 2.65 and 95th percentile = 3.39,
minimum = 1.96, maximum = 3.94; the normal distribution line is overlaid (Froese 2006).

12. b as Function of Size Range
Absolute residuals of b=3.0 plotted over the length range used for establishing the weight-length
relationship. The length range is expressed as fraction of the maximum length known for the
species. A robust regression analysis of absolute residuals vs. fraction of maximum length resulted
in n = 2,800, r2 = , slope = , 95% CL – (Froese 2006).

13. Mean b as a Function of Studies
Absolute residuals of mean b from b = 3.0, plotted over the respective number of
weight-length estimates contributing to mean b, for 1,773 species. The two outliers with
about 10 weight-length estimates belong to species with truly allometric growth (Froese 2006).

14. Understanding b
b ≈ 3
Isometric growth and small specimens have same condition as large specimens. Default.
b << 3
Negative allometric growth or small specimens are in better condition than large ones.
b >> 3
Positive allometric growth or large specimens are in better condition than small ones.

15. Understanding a If b ≈ 3 then a is a form-factor with
a ≈ > eel-like (eel)
a ≈ 0.01-> fusiform (cod, tunas)
a ≈ 0.1-> spherical (puffers, boxfish)

16. Understanding a
Frequency distribution of mean log a based on 3,929 records for 1,773 species, with
median a = , 95% CL = – , 5th percentile = ,
95th percentile = , minimum = , and maximum = (Froese 2006).

17. Interdependence of a and b
Any increase in slope b will decrease intercept a, and vice-versa.

18. log a vs. b Plot
Plot of log a over b for 25 length-weight relationships of Oncorhynchus gilae. The black dot
was identified as outlier by robust regression analysis (robust weight = 0.000).

19. Multi-species comparison
Scatter plot of mean log a (TL) over mean b for 1,232 species with body shape information.
Areas of negative allometric, isometric and positive allometric change in body weight relative to
body length are indicated. The regression line is based on robust regression analysis for fusiform
Species (Froese 2006).

20. More Information
Froese, R., Cube law, condition factor, and weight-length relationships: history, meta-analysis and recommendations. Journal of Applied Ichthyology 22(4): (915 Web of Science citations and 1638 Google Scholar citations in November 2017)
Froese, R., J.T. Thorson and R.B. Reyes, Jr A Bayesian approach for estimating length-weight relationships in fishes. Journal of Applied Ichthyology, 30(1):78-85 (80 Web of Science citations and 123 Google Scholar citations in November 2017)

21. Exercise In FishBase, find a species with at least 5 LWR studies
Discuss the variability of a and b
Select a study that describes LWR well and justify your selection
Find a species that deviates consistently from b ≈ 3 and discuss possible reasons