Length–Weight Relationships in Fish: What They Tell Scientists

What the length–weight relationship describes

A length–weight relationship is a species- or stock-specific curve that predicts the weight of a fish from its length. It is defined by two fitted parameters: a scaling coefficient and an exponent that describes how quickly weight increases as fish grow longer. By convention, and in FishBase, weight is expressed in grams and length in centimetres (FishBase manual).

Fisheries scientists use the relationship for three main purposes:

• Converting lengths to weights. Surveys and market sampling measure far more fish than they weigh, so length frequencies are turned into biomass with published parameters.
• Describing condition. Comparing observed weights with the weight expected at a given length shows whether fish are heavier or lighter than usual.
• Comparing populations. Differences in the parameters between areas, seasons or years can reflect growth, feeding or reproductive state.

Parameters only apply to the length type used to fit them. FishBase notes that published relationships based on fork length or standard length can be difficult to use with total length data, and it keeps a separate table of length–length conversions for that reason.

Why weight rises with the cube of length

Weight rises roughly with the cube of length because a fish that kept exactly the same shape and density as it grew would increase in height and width at the same rate as in length, so its volume would grow in all three directions at once. Scientists call this isometric growth, and it corresponds to an exponent of 3.

Change in length	Expected change in weight (isometric growth)
2% longer	about 6% heavier
10% longer	about 33% heavier
50% longer	about 3.4 times as heavy
Twice as long	about 8 times as heavy

Real fish rarely keep exactly the same shape, so the fitted exponent differs slightly from 3:

Exponent	Growth type	What it means
Exactly 3	Isometric	Shape does not change with size
Above 3	Positive allometric	Larger fish become relatively deeper or plumper
Below 3	Negative allometric	Larger fish become relatively more slender

In a meta-analysis of 3,929 relationships for 1,773 species, Froese (2006) confirmed that the exponent is expected to lie between 2.5 and 3.5, with a median across species of 3.03. The same study showed that plotting the two parameters of all published relationships for one species against each other helps to detect outliers. A fitted exponent outside the expected range is a reason to check the data: a narrow length range, mixed length types, unsorted sexes or weighing errors are common causes.

How scientists fit the relationship

The standard way to estimate the parameters is a straight-line regression on logarithmic scales, because the curved relationship between length and weight becomes a straight line when both are plotted on log axes. The slope of that line is the exponent. FishBase reports that ordinary regression of log weight on log length is used for the vast majority of published relationships.

1. Collect paired data. Measure and weigh individual fish across the full length range of the population, recording length type, sex, maturity, date and area.
2. Plot the raw data. Plot weight against length and remove obvious recording errors, such as a weight entered in kilograms instead of grams or a fish linked to the wrong length.
3. Transform and fit. Convert lengths and weights to logarithms and fit a straight line; the slope gives the exponent and the intercept gives the scaling coefficient.
4. Convert back with care. Predictions made on the log scale come out slightly too low when converted back to grams. A standard correction based on the scatter around the fitted line removes this bias (Sprugel, 1983).
5. Report fully. Publish the number of fish, length range, length type, units, goodness of fit and the period and area sampled, so others can judge whether the parameters apply to their data.

Relationships should not be used to predict weights far outside the length range on which they were fitted.

Worked example: cod around the British Isles

Using published parameters for Atlantic cod, a 50 cm fish is expected to weigh about 1.28 kg. Silva et al. (2013) fitted relationships to Cefas survey data from 2009–2012, with total length in centimetres and total weight in grams.

Dataset (Silva et al., 2013)	Fish measured	Length range (cm)	Coefficient	Exponent	r²
Cod, British Isles (all surveys)	2,073	8–119	0.0098	3.0109	0.9952
Cod, North Sea groundfish survey (Q3)	1,085	16–111	0.0081	3.0502	0.9929

Entering these parameters in a spreadsheet or statistics package gives the predicted weights below.

Total length	Predicted weight, British Isles parameters	Predicted weight, North Sea Q3 parameters
30 cm	275 g	259 g
50 cm	1,278 g	1,232 g
80 cm	5,263 g	5,168 g

Two points stand out. First, the two parameter sets differ by 2–6% in predicted weight over this range, a reminder that relationships vary between areas and seasons. Second, because weight follows roughly the cube of length, a 1 cm error in length at 50 cm changes the predicted weight by roughly 6%. That is why length conventions such as "to the centimetre below" must be applied consistently.

Condition factor and relative weight

A condition factor compares a fish's weight with the weight expected for its length, showing whether it is heavier or lighter than a typical fish of that size. Fulton's condition factor, the classic version, relates weight in grams to the cube of length in centimetres and is scaled so that values for many species lie around 1. A 50 cm cod weighing 1,300 g scores about 1.04, slightly above the value predicted for that length.

Fulton's factor is easiest to interpret when the species grows close to isometrically. When the exponent departs from 3, the factor drifts with length even for fish in identical condition, so a large fish may look fatter or thinner than a small one purely because of its size. Froese (2006) therefore recommended relative weight, the observed weight expressed as a percentage of the weight predicted from a species-wide relationship, for comparing individuals across populations.

Measure	Compares	Best used for
Fulton's condition factor	Weight with the cube of length	Fish of similar length within one stock
Relative weight	Weight with the weight predicted for that length	Comparisons across sizes and populations

Why precise weights at sea matter

For small fish, the scale resolution becomes a large fraction of the fish's weight, which adds noise to the relationship and weakens every estimate derived from it. The table uses species parameters from Silva et al. (2013) to show how large a single display step is relative to the predicted weight.

Fish and length	Predicted weight	1 g step as % of weight	0.1 g step as % of weight
Sprat, 8 cm	3.8 g	26%	2.6%
Sprat, 10 cm	7.3 g	14%	1.4%
Herring, 20 cm	63 g	1.6%	0.16%
Cod, 30 cm	275 g	0.36%	0.04%

The same report shows the consequence in real data: sprat recorded to 0.5 cm and 1 g gave a goodness of fit (r²) of 0.70, whereas sprat recorded to 0.1 cm and 0.1 g gave 0.865, and the authors recommended that surveys reconsider the resolution used for such species.

Motion adds a second source of noise. On a moving vessel, vertical accelerations change the force on the load cell, so a scale without motion compensation shows a fluctuating value and operators tend to record whatever number is displayed at the moment they look. For small fish this noise can exceed the resolution itself. Why these effects are hardest for small samples is explained in Precision Weighing of Small Samples at Sea, and the sampling context in Biological Sampling on Research Vessels.

Length–weight relationships are also published for benthic invertebrates, for example 216 North Sea species by Robinson et al. (2010), and the same resolution arguments apply to small crustaceans and molluscs.

How WPL approaches this

For individual fish weights, WPL's M3 Series Scientific Marine Scale is motion-compensated and offers 300 g (0.1–0.2 g readability) and 600 g (0.2–0.5 g) ranges for small fish such as sprat and juvenile gadoids, and 1,500 g to 6,000 g ranges for larger fish on a 270 × 270 mm platform. Weights can be logged automatically with WeightControl, avoiding transcription errors. For the wider context, see the Research Vessels hub.