For most fisheries applications, the shape of a length-frequency distribution is much more important than its mean length or variance. This makes it difficult to evaluate at which point a sample size is adequate. By estimating the coefficient of variation of the counts in each length

class and taking a weighted mean of these, a measure of precision was obtained that takes the precision in all length classes into account. The precision estimates were closely associated with the ratio of the sample size to the number of size classes in each sample. As a rule-of-thumb, a

minimum sample size of 10 times the number of length classes in the sample is suggested because the precision

deteriorates rapidly for smaller sample sizes. In absence of such a rule-of-thumb, samplers have previously

under-estimated the required sample size for samples with large fish, while over-sampling small fish of the same species

Gerritsen, Hans D.

McGrath, David

Aquatic Commons

116
ART & EQUATIONS ARE LINKED
Length measurements are fundamen-
tal to many aspects of ﬁsheries sci-
ence. However, there is little formal 
guidance on the appropriate size of 
a length sample. Such guidance is of 
particular relevance when the number 
of ﬁsh available exceeds the number 
that can be measured at a reasonable 
cost, and a subsample needs to taken. 
Clearly, the required precision of a 
length sample depends on the purpose 
of sampling. In order to identify modes 
of individual year classes for a length-
based assessment, the precision of the 
sample needs to be quite high. Sample 
sizes of more than 1000 are neces-
sary to identify more than half the 
modes in a typical length distribution 
(Erzini, 1990). A sample size of at 
least 100 adult ﬁsh was recommended 
for age-based stock assessment pur-
poses (Anderson and Neumann, 1996), 
although the authors did not mention 
how they arrived at this number. 
Regardless of the type of assess-
ment that is used, the shape of the 
length-frequency distribution is of 
interest, rather than simple sum-
mary statistics such as the mean 
or the variance. For this reason, it 
has proved difﬁcult to quantify what 
constitutes a representative or ad-
equately precise length distribution. 
Some studies have attempted to ﬁnd 
minimum or optimum sample sizes 
by comparing samples to an expected 
distribution (e.g., Muller1; Gomez-
Buckley et al.2; Vokoun et al., 2001). 
However, the true distribution is 
usually unknown, and dissimilarity 
Precision estimates and suggested sample sizes  
for length-frequency data
Hans D. Gerritsen (contact author)1
David McGrath2
Email address (for H. D. Gerritsen): hans.gerritsen@marine.ie
1 Fisheries Science Service, Marine Institute
 Rinville, Oranmore
 County Galway, Ireland
2 Commercial Fisheries Research Group
 Galway-Mayo Institute of Technology
 Dublin Road
 Galway, Ireland. 
Manuscript submitted 17 May 2006  
to the Scientiﬁc Editor’s Ofﬁce.
Manuscript approved for publication  
22 June 2006 by the Scientiﬁc Editor.
Fish. Bull. 106:116–120 (2007). 
Abstract—For most fisheries applica-
tions, the shape of a length-frequency 
distribution is much more important 
than its mean length or variance. 
This makes it difficult to evaluate 
at which point a sample size is ade-
quate. By estimating the coefficient of 
variation of the counts in each length 
class and taking a weighted mean 
of these, a measure of precision was 
obtained that takes the precision in 
all length classes into account. The 
precision estimates were closely asso-
ciated with the ratio of the sample 
size to the number of size classes in 
each sample. As a rule-of-thumb, a 
minimum sample size of 10 times 
the number of length classes in the 
sample is suggested because the preci-
sion deteriorates rapidly for smaller 
sample sizes. In absence of such a 
rule-of-thumb, samplers have previ-
ously under-estimated the required 
sample size for samples with large 
fish, while over-sampling small fish 
of the same species.
from the expected distribution does 
not necessarily indicate an imprecise 
sample. In addition, these methods 
provide only indirect measures of pre-
cision that are difﬁcult to evaluate 
objectively. 
Thompson (1987) used the precision 
of a sample explicitly to establish an 
appropriate sample size. Thompson 
proved that a sample size of 510 is 
sufﬁcient to be 95% conﬁdent that all 
estimated proportions in a multinomi-
al distribution are no more than 5% 
from the true proportion. However, 
Thompson based this ﬁgure (n=51) on 
a worst-case scenario, which, in the 
present case, is a length-frequency 
distribution that is evenly apportioned 
over three size classes. Because this 
is not the typical shape of a length-
frequency distribution used in ﬁsher-
ies science, Thompson’s measure of 
precision is too conservative for the 
vast majority of cases.
For most ﬁsheries applications, it 
would be more useful to define the 
1 Müller, H. 1996. Minimum sample 
sizes for length distributions of the catch 
estimated by an empirical approach. 
ICES CM 1996/J12, 18 p.
2 Gomez-Buckley, M., L. Conquest, S. 
Zitzer, and B. Miller. 1999. Use of 
statistical bootstrapping for sample size 
determination to estimate length-fre-
quency distributions for pacific albacore 
tuna (Thunnus alalunga). Final report 
to National Marine Fisheries Services, 
FRI UW 9902, 7 p. Website: http://www.
fish.washington.edu/research/publica-
tions/pdfs/9902.pdf (accessed 31 March 
2006).
117Gerritsen and McGrath: Precision estimates and suggested sample sizes for length-frequency data
PREFLIGHT GOOD TO GO
precision of a length-frequency sample as the mean 
precision over the entire size range. However, it appears 
that this approach has not been used to establish an op-
timum sample size. Such mean precision estimates over 
the entire size range might be used to obtain a rule-
of-thumb for sample sizes that are required in order 
to obtain a certain precision level of the catch at each 
location. In the present study we aim 1) to determine a 
rule-of-thumb for obtaining an appropriate sample size 
when the number of ﬁsh available in a particular sample 
exceeds the number that can be measured at a reason-
able cost, and 2) to examine the sample sizes that have 
been taken in the past, in absence of such guidance.
Materials and methods
Data were used from the Irish Groundﬁsh survey, which 
was carried out on RV Celtic Explorer in the waters 
around Ireland during October and November 2005. 
The catch was sorted into species and, if appropriate, 
into size grades, each of which were treated as a sepa-
rate length sample. Length measurements were taken 
from all ﬁsh and squid species that were caught. If the 
number of individuals in a sample was large, a sub-
sample was taken by repeatedly transfering the sample 
from each ﬁsh box into two other boxes and discarding 
one of these. This method ensures that the entire catch 
is represented uniformly in the subsample. At the time 
of the survey, the samplers did not have any particular 
guidance on the appropriate size for a subsample; they 
used their own judgment to decide on the sample size.
The precision of the number of observations in each 
length class of a random sample can be estimated by as-
suming a multinomial distribution (Smith and Maguire, 
1983). If the precision in each length class is expressed 
in the form of a coefﬁcient of variation (CV), an overall 
measure of precision can be obtained by weighting each 
CV by the number of ﬁsh in each length class. This 
mean weighted CV (MWCV) provides a description of 
the precision over the entire range of size classes in a 
length frequency distribution. 
Under the assumption of a multinomial distribution, 
the standard deviation (σi) of the number of ﬁsh in a 
sample that are length category i can be estimated by
 σ i i inp p= −( ),1  (1)
where n = the total number of ﬁsh in the sample; and 
 pi = the proportion of the sample that is length i. 
The coefﬁcient of variation (CV) of the number of ﬁsh at 
length i, is given by
 CV
npi
i
i
= σ  (2)
and the mean weighted coefﬁcient of variation (MWCV) 
is given by
 MWCV p CV
ni i
i= =∑ ∑σ .  (3)
The highest possible value of the MWCV results from a 
length-frequency distribution that is evenly distributed 
over a large number of size classes. The number of ﬁsh 
at each length class are then Poisson distributed with 
a standard deviation that equals the square root of the 
number at length (Zar, 1999). The theoretical maximum 
MWCV is therefore given by
 MWCV n c= −( / ) ,.0 5  (4)
where c =  the number of size classes in the sample. 
The minimum MWCV is zero and would result from a 
distribution where all observations fall within a single 
length category. Therefore, the MWCV estimates will 
always lie between zero and the curve described by 
Equation 4.
Results
During the 2005 survey, a total of 2332 length samples 
were taken for 80 different species of ﬁsh and squid. In 
most cases, the sample size was limited by the number 
of individuals in the catch. However, 596 samples were 
deemed too large to measure all individuals and sub-
samples were taken. The median subsample size was just 
under a quarter of the total catch (by weight), whereas 
90% of the subsamples were smaller than half of the total 
catch. The four most common species that were subsam-
pled were poor cod (Trisopterus minutus), blue whiting 
(Micromesistius poutassou), haddock (Melanogrammus 
aegleﬁnus), and Norway pout (Trisopterus esmarkii).
The estimated MWCV of the subsamples was closely 
associated with the ratio of the number of individuals 
measured to the number of length classes in the sample 
(Fig. 1). The MWCV appeared to follow an exponential 
curve that was close to the maximum MWCV given by 
Equation 4. The MWCV decreased very rapidly with 
increasing sample size up to sample sizes of around 
10 times the number of length classes in the sample, 
after which the sample size would need to be increased 
considerably for a moderate further improvement in 
precision. If the sample size is taken as 10 times the 
number of length classes in the distribution, an MWCV 
of around 0.25 can be expected; a sample size of 48 
times the number of length classes would result in 
an MWCV of 0.10 and a sample size of 155 times the 
number of length classes would be necessary to reduce 
the MWCV to 0.05. 
The mean sample size in the subsamples taken on 
the survey was just under nine times the number of 
length classes per sample, resulting in a mean MWCV 
of 0.33. However, there was quite a large spread in 
the sample sizes (Fig. 1); therefore some samples were 
measured with very low precision, whereas others had 
118 Fishery Bulletin 105(1)
Figure 1
The mean weighted coefficient of variation (MWCV) 
for 596 subsamples was closely related to the sample 
size (n) divided by the number of per length classes in 
the sample (c). A good fit was obtained for the power 
function indicated by the solid line; its parameters 
are given at the top of the plot. The dashed line 
indicates the theoretical maximum MWCV (Eq. 4.). 
The histograms show the distribution of the samples 
on both axes. 
M
W
C
V
Sample size per length class (n/c)
1.0
0.8
0.6
0.4
0.2
0.0
0 10 20 30 40
MWCV = 0.98(n/c)–0.59
r 2 = 0.98
excessively large sample sizes. The range of sample 
sizes was between 2.2 and 24.7 times the number of 
length classes (2.5% and 97.5% quantiles), resulting 
in a range of MWCVs between 0.14 and 0.61. With 
a minor increase in effort, the sample size might be 
increased to 10 per length class for each subsample, 
resulting in an MWCV of around 0.25 for all samples. 
Considering that the precision deteriorates very rap-
idly for sample sizes of less than 10 per length class, a 
minimum sample size of 10 times the number length 
classes in the sample is suggested as a rule-of-thumb 
in the present case. 
The previous analysis shows that, in order to ob-
tain the same level of precision for all subsamples, 
the sample size should be directly proportional to the 
number of size classes. In absence of specific guid-
ance on the sample size during the 2005 survey, the 
chosen sample size was only weakly correlated to the 
number of length classes in the sample of poor cod 
and haddock, whereas no significant correlation was 
found for blue whiting and Norway pout (Fig. 2). The 
same figure also shows that the MWCV in subsamples 
tended to increase with the mean length of the fish in 
the sample. This increase indicates that samples with 
a large mean size tended to be sampled with lower 
precision than samples of smaller fish of the same 
species. 
Discussion
Length distributions that result from combining a 
number of different samples exhibit greater variation 
than predicted under the multinomial model given in 
Equation 1 (Smith and Maguire, 1983). Fish populations 
are usually not uniformly mixed; therefore individual 
samples are not random samples from the population 
(Pennington et al., 2002). The simple multinomial model 
does not take account of the between-sample variability 
and will therefore underestimate the total variance. 
However, Equation 1 does provide an unbiased estimate 
of the variability within each sample, which is the vari-
ability that would occur if one could repeatedly take 
a random sample at the same location and time and 
measure these without error. This is the variability that 
is of interest when deciding whether the sample size is 
large enough to estimate the length distribution from a 
particular haul with a certain precision. Therefore, the 
MWCV is a suitable measure for this exercise.
In order to obtain a precise population estimate, it 
is important to maximize the number of sampling lo-
cations because of the considerable between-sample 
variability that is usually present (Pennington et al., 
2002). Pennington et al. (2002) suggested maximiz-
ing the number of sampling locations at the expense 
of the number of ﬁsh measured. However, the number 
of hauls is often limited by practical considerations, 
and length measurements can be obtained quickly and 
cheaply. Therefore, it seems prudent to sample enough 
ﬁsh from each haul to obtain a length distribution that 
is representative of that catch at that particular loca-
tion. Detailed information on the length distribution 
at each station can be valuable for exploratory data 
analysis, such as investigating the spatial structure in 
the data. Nevertheless, this level of sampling may not 
be strictly necessary for a precise population estimate of 
the length-frequency distribution for an age- or length-
based assessment.
The samples in Figure 1 included a large range of 
species and size categories of ﬁsh, but the variability in 
the MWCV was small after taking account the sample 
sizes. This small amount of variability indicates that 
the MWCV is not very sensitive to the exact shape of 
the distribution and can be predicted with high preci-
sion, at least within the range of distributions encoun-
tered on the survey. A minimum sample size of 10 times 
the number of length classes in the sample appears to 
be a reasonable compromise between effort and preci-
sion in the present case. 
The current analysis has focused on subsampling 
during surveys; however the same principles can be 
applied to any process of collecting data for which the 
shape of the distribution is of interest. The desired pre-
cision level for these cases will depend on a number of 
factors. For certain species that are of little commercial 
or scientiﬁc interest, but which may span across a large 
number of length classes, the suggested sample size of 
10 per length class may be excessive. Likewise, as the 
MWCV is directly proportional to the number of length 
119Gerritsen and McGrath: Precision estimates and suggested sample sizes for length-frequency data
classes in the sample, the choice of the interval of the 
length classes will determine the precision. Although 
increasing the size of length intervals will reduce the 
MWCV, this action will result in a loss of information 
which is undesirable. The cost of sampling, the detail 
required, and the purpose of the data collection need 
to be considered before the required precision level can 
be determined for applications other than the present 
example.
Without formal guidance on the appropriate sample 
size, the sample sizes chosen were, at best, weakly 
correlated with the number of size classes in the sam-
ples. It appears that the samplers under-estimated 
the required sample size for samples with large ﬁsh, 
whereas samples of smaller ﬁsh of the same species 
were over-sampled. This tendency to under-estimate the 
sample size may be related to the fact that the volume 
of a sample increases with the cube of its mean length; 
therefore a sample size of large ﬁsh may appear to be 
larger than the same number of small ﬁsh. In addition, 
samples with large ﬁsh tend to be spread out over a 
larger number of size classes, thus requiring higher 
sample numbers.
In practice, it will be difﬁcult for a sampler to esti-
mate both the number of size classes and the number 
of ﬁsh in a sample. Therefore, the Marine Institute in 
Ireland is developing a software application that al-
lows samplers to examine the length frequencies of the 
samples directly after they have been measured. The 
software estimates the weight of the suggested sample 
size for each distribution. Because size distributions 
tend to be similar on consecutive hauls, the sampler can 
gain an insight into the required weight of an appropri-
ate sample for each species and size category. 
The information contained in a length-frequency dis-
tribution is largely a function of sample size. The pres-
ent method allows the amount of information contained 
in a length-frequency distribtuion to be quantiﬁed in 
terms of precision, allowing samplers to make informed 
decisions on the sample size that is required to obtain 
an adequate estimate of the length-frequency distribu-
tion of a particular catch.
S
am
pl
e 
si
ze
Number of length classes
Mean length (cm)
M
W
C
V
300
200
100
0
0
15 14 18 22 2620 25 30 35 10 8 10 14 1812 14 16 18
5 10 20 30 0 5 10 20 30 0 5 10 20 30 0 5 10 20 30
0.6
0.4
0.2
0.0
Haddock Blue whiting Norway pout Poor cod
Haddock Blue whiting Norway pout Poor cod
r 2=0.27; P<0.001 r 2=0.05; P=0.05 r 2=0.00; P=0.88 r 2=0.06; P=0.03
r 2=0.67; P<0.001 r 2=0.29; P<0.001 r 2=0.66; P<0.001 r 2=0.43; P<0.001
Figure 2
The sample sizes of subsamples taken on the survey were correlated with the number of length classes in 
the samples of haddock (Melanogrammus aeglefinus) and poor cod (Trisopterus minutus), but not signifi-
cantly so for blue whiting (Micromesistius poutassou) and Norway pout (Trisopterus esmarkii) (top row). 
There was considerable variation in the mean weighted coefficient of variation (MWCV), which correlated 
with the mean length of fish in the samples (bottom row). The solid lines represent linear regressions and 
the dashed lines indicate the sample sizes and MWCV that would have resulted from a sampling scheme 
where the sample size was chosen to be 10 times the number of length classes in the distribution. The 
coefficients of determination, r2, are given together with their P-values.
120 Fishery Bulletin 105(1)
Acknowledgments
Free R-software has been used for this work and the 
authors would like to thank the R development core 
team and all contributors to the R project (http://www.
R-project.org). We also thank Colm Lordan and two 
anonymous reviewers for their valuable comments.
Literature cited
Anderson, R. O., and R. M. Neumann. 
1996. Length, weight, and associated structural 
indices. In Fisheries techniques, 2nd ed. (B. R. Murphy 
and D. W. Willis, eds.), p. 447−482. Am. Fish. Soc., 
Bethesda, MD.
Erzini, K. 
1990. Sample size and grouping of data for length-fre-
quency analysis. Fish. Res. 9:355−366.
Pennington, M., L. M. Burmeister, and V. Hjellvik. 
2002. Assessing the precision of frequency distribu-
tions estimated from trawl-survey samples. Fish. 
Bull. 100:74−80.
Smith, S. J., and J. J. Maguire. 
1983. Estimating the variance of length composition 
samples. In Sampling commercial catches of marine 
fish and invertebrates (W. G. Doubleday and D. Rivard, 
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Thompson, S. K. 
1987. Sample size for estimating multinomial propor-
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665.
Zar, J. H.
1999. Biostatistical analyis, 4th ed., 663 p. Prentice-
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Precision estimates and suggested sample sizes for length-frequency data

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Precision estimates and suggested sample sizes for length-frequency data

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