Determining Optimum Quadrat Size and Shape: An Introduction

Determining Optimum Quadrat Size and Shape:
An Introduction
INTRODUCTION
Quadrat sampling techniques are among the oldest techniques in ecology. Counts are simple
to comprehend and can be used in a great variety of ways on organisms as diverse as trees,
barnacles, kangaroos, seabirds, and caribou. There are only 2 basic requirements of all the
techniques: 1) the area (or volume) counted is known, and 2) the organisms are relatively
immobile during the counting period.
The term quadrat strictly means a 4-sided figure, but in practice this term is used to mean
any sampling unit, whether circular, hexagonal, or even irregular in outline, which can be
placed on the ground so that % plant cover can be estimated, plants and animals counted,
species listed, etc.
Quadrat counts have been used extensively on plants, which rarely run away while being
counted, but they are also suitable for animals if the person counting is keenly observant or
has a camera. This problem set introduces the process of determining the optimum quadrat
size and shape for use in monitoring programs that are estimating the abundance of plants
and animals.
PROCEDURES
The details of quadrat sampling, including plot size, shape, number, and arrangement of
sample plots, must be determined for the particular type of community being sampled and
on the basis of the type of information desired. For good discussions of designing quadrat
sampling methodology, see Goldsmith et al. (1986) and Krebs (1999).
Quadrat size and shape
If you are going to sample a forest community, say to estimate the abundance of snags, you
must first make 2 operational decisions: 1) what size of quadrat should I use? and 2) what
shape of quadrat is best? Answers to these questions can be complex.
Two approaches have been used to determine quadrat size and shape. The simplest approach
is to go to the literature and use the same quadrat size and shape that everyone else uses.
Thus, if you are sampling snags in a mature forest, you will find that most people use
quadrats 10m x 10m (100 m 2 ); for herbaceous vegetation, most use 0.1-1.0 m 2 -sized plots;
and for shrubs or saplings ≤3 m tall, 10-20 m 2 -sized plots are used most commonly. The

problem with this approach is that the accumulated wisdom of ecologists is not yet sufficient
to assure you of the correct answer.
A better approach, if time and resources are available, is to determine for your particular
study the optimal quadrat size and shape. To do this, we first must decide on what we mean
by “best” or “optimal.” Best can be defined in 3 ways: 1) statistically, as that quadrat size and
shape giving the highest statistical precision for a given total area sampled, or for a given total
amount of time or money; 2) ecologically, as that quadrat size and shape that are most
efficient to answer the question being asked; and 3) logistically, as that quadrat size and shape
that are easiest to establish in the field and use. Be wary of the logistical criterion, however,
since in many ecological cases the easiest approach is rarely the best. If you are investigating
questions of ecological scale, the processes you are studying will dictate quadrat size, but in
most cases the statistical criterion and the ecological criterion are the same. In all these cases
we define
highest statistical precision = lowest standard error of the mean
= narrowest confidence interval for the mean.
In almost all cases we will encounter, we should determine the quadrat size and shape that will
give us the highest statistical precision. How can we do this?
Let’s consider the shape question first. There are 2 conflicting problems with shape. First, the
edge effect is smallest in a circular quadrat, and largest in a rectangular one. The ratio of
length of edge to the area inside a quadrat changes as follows:
Circle < Square < Rectangle
Edge effect is important because it leads to possible counting errors. A decision must be
made every time an animal or plant is at the edge of a quadrat: Is this individual inside or
outside the area to be counted? This decision is often biased because most biologists prefer
to count an organism rather than ignore it. Edge effects thus often produce positive biases
(i.e., the numbers of organisms counted within the quadrat are higher than they should be,
which could obviously lead to a population estimate that is too high). Proportionally, the
more edge your plot has the more positive bias results. The general significance of possible
errors of counting at the edge of a quadrat cannot be quantified because it is organism- and
habitat-specific. It can, however, be reduced by training. If edge effects are a significant
source of error, you should choose a quadrat shape with less edge/area. But how do you
know you might have an edge effect problem? Figure 1 illustrates a way of recognizing an
edge effect problem. Note that there is no reason to expect any bias in mean abundance
estimated from a variety of quadrat sizes and shapes. If there is no edge effect bias, we expect
in an ideal world to get the same mean value regardless of the size or shape of the quadrats
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used, if the mean is expressed in the same units of area. This is important to remember:
Quadrat size and shape are not about biased abundance estimates but are about narrower
confidence limits. If you find a relationship like that in Figure 1 in a data set you are working
on, you should immediately disqualify the smallest quadrat size from consideration to avoid
bias from the edge effect.
Figure 1. Edge effect bias in small quadrats. The
estimated mean dry weight of grass (per 0.25
m 2 ) is much higher in quadrat size 1 (0.016 m 2 )
than in all other quadrat sizes. This suggests an
overestimation bias due to edge effects, and that
quadrat size 1 should not be used to estimate
abundance of these grasses. The data are from
Wiegert (1962).
The second problem regarding quadrat shape is that nearly everyone has found that long,
thin quadrats are better than circular or square ones of the same area. The reason for this is
because of habitat heterogeneity. Long quadrats cross more patches and thus capture more
of the habitat variability (something you want to do). Habitats are rarely uniform, and
organisms are usually distributed somewhat patchily within the overall sampling zone.
Clapham (1932) counted the number of self-heal (Prunella vulgaris) plants in 1-m 2 quadrats
of 2 shapes: 1m x 1m (square) and 4m x 0.25 m (rectangle). He counted 16 quadrats and got
these results:
Quadrat size Mean Variance SE 95% CI
1m x 1m 24 565.3 5.94 ±12.65
4m x 0.25m 24 333.3 4.56 ±9.71
Clearly in this situation, the rectangular quadrats are more efficient than square ones (i.e., the
SE is lower and the CI is narrower). Given that only 2 shapes of quadrats were tried, we do
not know if even longer, thinner quadrats might be still more efficient.
Not all sampling data show this preference for long, thin quadrats, and for this reason each
situation should be analyzed independently. Table 1 shows data from basal area
measurements on trees in a forest stand studied by Bormann (1953). The observed SD almost
always falls as quadrat area increases. But if an equal total area is being sampled, the highest
precision (=lowest SE) will be obtained by using 70 4m x 4m quadrats (Table 1). If, on the
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other hand, an equal number of quadrats were to be taken for each quadrat size, one would
prefer the long, thin quadrat shape.
Table 1. Effect of quadrat size on SD for measurements of basal area of trees in an oakhickory
forest in North Carolina. The data are from Bormann (1953).
Quadrat size Observed SD
(m) per 4 m2 Sample sizea SE of the
mean for
sample size
4 x 4 50.7 70 6.06
4 x 10 47.3 28 8.94
4 x 20 44.6 14 11.92
4 x 70 41.3 4 20.65
4 x 140 34.8 2 24.61
aThe number of quadrats of a given size needed to sample 1,120 m2 .
Two methods are available for choosing the best quadrat size statistically. Wiegert (1962)
proposed a general method that can be used to determine optimal size or shape. Hendricks
(1956) proposed a more restrictive method for estimating optimal size of quadrats. In both
methods it is essential that data from all quadrats be standardized to a single unit area—for
example, per m2 . This conversion is simple for means, SDs, and SEs: divide by the relative
area. For example,
Mean number per m2 2
Mean number per 0.25m
=
0.25
SD per m2 2
SD4m
=
4
For variances, the square of the conversion factor is used:
Variance per m2 Variance per 9m
= 2
9
2
For both Wiegert’s and Hendrick’s methods, you should standardize all data to a common
base area before testing for optimal size or shape of quadrat. They both assume further that
you have tested for and eliminated quadrat sizes that give an edge effect bias. We will only
deal with Wiegert’s method here.
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Wiegert’s method of determining optimum quadrat size and shape
Wiegert (1962) proposed that 2 factors were of primary importance in deciding on optimal
quadrat size or shape: relative variability and relative cost. In any field study, time or money
would seem to be the limiting resource, and we must consider how to optimize with respect
to sampling time. We will assume that time = money, and in the formulas that follow, either
unit may be used. Costs of sampling have 2 components (in a simple world anyway):
C = C0 + Cx,
where C = Total cost for 1 sample
C0 = Fixed costs or overhead
Cx = Cost for taking 1 sample quadrat of size x.
Fixed costs involve the time spent walking or flying between sampling points and the time
spent locating a random point for the quadrat; these costs may be trivial in an open grassland;
they can be enormous when sampling the ocean in a large ship. The cost for taking a single
quadrat may or may not vary with the size of the quadrat. Consider a simple example from
Wiegert (1962) of grass biomass in quadrats of different sizes:
Quadrat size (area)
1 3 4 12 16
Fixed cost ($) 10 10 10 10 10
Cost per sample ($) 2 6 8 24 32
Total cost for 1 quadrat ($) 12 16 18 34 42
Relative cost for 1 quadrat 1 1.33 1.50 2.83 3.50
We need to balance these costs against the relative variability of samples taken with quadrats
of different sizes:
Observed variance per 0.25
m 2
Quadrat size (area)
1 3 4 12 16
0.97 0.24 0.32 0.14 0.15
The operational rule is, Pick the quadrat size that minimizes the product of (Relative cost) x
(Relative variability).
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In this example, we begin by disqualifying quadrat size 1 because it showed a strong edge
effect bias (Figure 1). Figure 2 shows that the optimal quadrat size for grass in this particular
study is 3, although there is relatively little difference between the products for quadrats of
size 3, 4, 12, and 16. In this case, the size 3 quadrat gives the maximum precision for the least
cost.
Figure 2. Determining the optimal quadrat size
for sampling. The best quadrat size is that which
gives the minimal value of the product of relative
cost and relative variability, which is quadrat size
3 in this example. Quadrat size 1 is plotted here
for illustration, even though this quadrat size is
disqualified because of edge effects. Data here are
dry weights of grasses in quadrats of variable size.
Data are from Wiegert (1962).
Let’s work through an example from start to finish. The data we are using comes from
Pringle’s (1984) study of seaweed (Chondrus crispus) biomass. As you can see from Table 2,
Pringle tried quadrats of several different sizes in a pilot study to assist him in selecting the
most appropriate-sized quadrat.
Table 2. Data to determine optimal quadrat size for biomass estimates of the seaweed,
Chondrus crispus (from Pringle 1984).
Quadrat size
(m)
Sample size a
Mean
biomass b
(g)
SD b
SE of the
mean b
Time to
take 1
sample
(min)
0.5 x 0.5 79 1524 1022 115 6.7
1 x 1 20 1314 963 215 12.0
1.25 x 1.25 13 1037 605 168 13.2
1.5 x 1.5 9 1116 588 196 11.4
1.73 x 1.73 6 2021 800 327 33.0
2 x 2 5 1099 820 367 23.0
a Sample sizes are approximately the number to make a total sampling area of 20 m 2 .
b All expressed per m 2 .
If we neglect any fixed costs and use the time to take 1 sample as the total cost, we can
calculate the relative cost for each quadrat size as
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Relative cost =
Time to take 1 sample of a givensize
Minimum time to take 1 sample
Equation 1
In this case the minimum time = 6.7 minutes for the 0.5 x 0.5 quadrat (that shouldn’t be
surprising). We can also express the variance [=(SD) 2 ] on a relative scale:
2
(SD)
Relative variance = 2
Equation 2
(Minimum SD)
In this case, the minimum variance occurs for quadrats of 1.5 x 1.5 m. We obtain for these
data:
Quadrat size
(m)
Relative
variance
Relative
cost
Product of
relative variance
and relative cost
0.5 x 0.5 3.02 1.00 3.02
1 x 1 2.68 1.79 4.80
1.25 x 1.25 1.06 1.97 2.09
1.5 x 1.5 1.00 1.70 1.70
1.73 x 1.73 1.85 4.93 9.12
2 x 2 1.94 3.43 6.65
The operational rule is to pick the quadrat size with minimal product of cost x variance, and
this is clearly the 1.5 x 1.5 m quadrat, which is the optimal quadrat size for this particular
sampling area.
There is a slight suggestion of a positive bias in the mean biomass estimates for the smallest
quadrats (Table 2), but this was not tested for by Pringle. Optimal quadrat shape could be
decided in exactly the same way as quadrat size.
Would there ever be a time when you might not want to determine optimum quadrat size
and shape? Sampling plant and animal populations with quadrats is done for many different
reasons, and it is important to ask when you might be advised to ignore the
recommendations for quadrat size and shape that arise from Wiegert’s method.
There are 2 common situations when you may wish to ignore these recommendations. In
some cases you will wish to compare your data with older data gathered with a specific
quadrat size and shape. Even if the older quadrat size and shape are inefficient, you may be
advised, for comparative purposes, to continue using the old quadrat size and shape. In
principle this is not required, as long as no bias is introduced by a change in quadrat size. But
in practice, many biologists are more comfortable using the same quadrats as the earlier
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studies. This can become a more serious problem in a long-term monitoring program in
which a poor choice of quadrat size in the early stages could condemn the whole project to
wasting time and resources with inefficient sampling procedures.
If you are sampling several habitats, sampling for many different species, and/or sampling
over several seasons, you may find that these procedures result in a recommendation for a
different size and shape of quadrat for each situation. It may be impossible to do this for
logistical reasons, and thus you may have to compromise.
NOW YOU DO IT
Exercise #1: A field plot was located in the subalpine zone of Rocky
Mountain National Park, Colorado and divided into 16 quadrats of 1-m 2 .
The numbers of American bistort, Polygonum bistortoides, were
counted in each quadrat with these results:
3 0 5 1
5 7 2 0
3 2 0 7
3 3 3 4
Calculate the precision (i.e., SE of the mean and CI) of sampling this universe with 2 possible
shapes of 4-m 2 quadrats: 2m x 2m and 4m x 1m.
Exercise #2: Using the mapped population of a 17,860-m 2 (1.8 ha) stand of hardwoods in
northern Minnesota provided, use Wiegert’s method to determine the optimum quadrat size
to determine the number of balsam fir trees (species #1 on the map). Table 3 contains the
quadrat sizes you will compare, the required sample sizes for each quadrat size (yes, you will
need to sample at the intensity indicated), and the estimated time required to take 1 sample
at each quadrat size (these numbers are provided and are mere guesses of the amount of time
it would actually take to search a quadrat of each size for balsam fir trees). Notice that
sample size is being held constant relative to the quadrat size so that each quadrat size is
sampling the same area. We will discuss ways of determining sample sizes latter. You need to
calculate the mean number of balsam fir trees, the SD, and the SE of the mean for each
quadrat size. Record your data in Table 3. With these data, use Equations 1 and 2 to
determine optimum quadrat size. Record your calculations in Table 4.
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Use the 3 quadrats on the overhead transparency provided. Quadrats could be located
randomly using the 2 numbered axes placed along the edges of the sampling frame. As you
can see, the axes divide the sampling frame into cells. Using a random numbers table, you
could select a pair of numbers (x and y) and use these numbers to place your quadrat on the
map. Perhaps you would place the lower left corner of each quadrat on each randomlylocated
point. Don’t forget to answer the 3 questions below.
Table 3. Optimal quadrat size for population estimates of balsam fir.
Quadrat size
(m)
Sample size a
Mean
number of
balsam fir b
SD b
SE of the
mean b
Time to
search 1
quadrat
(min)
10m x 10m 50 5
25m x 25m 8 30
50m x 50m 2 125
a Sample sizes are the number of quadrats to make a total sampling area of 5,000m 2 .
b All expressed per m 2 .
Table 4. Wiegert’s method to determine optimal quadrat size for counting balsam fir in a
northern hardwood stand.
Quadrat size
(m)
10m x 10m
25m x 25m
50m x 50m
QUESTIONS:
Relative
variance
Relative
cost
Product of
relative variance
and relative cost
1. What are the SE and CI for the 2 quadrate sizes used in exercise #1?
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2. What is the most optimum quadrat size to count balsam fir in this northern hardwood
stand?
3. How would you go about determining optimum quadrat shape? Work up an approach
and do it. A couple of blank overhead transparencies are provided for your use. What is
the most optimum quadrat shape to count balsam fir in this northern hardwood stand?
ACKNOWLEDGEMENTS
The bulk of this problem set was taken from Krebs (1999), particularly chapter 4.
LITERATURE CITED
Bormann, F.H. 1953. The statistical efficiency of sample plot size and shape in forest ecology.
Ecology 34:474-487.
Clapham, A. R. 1932. The form of the observational unit in quantitative ecology. Journal of
Ecology 20:192-197.
Goldsmith, F.B., C.M. Harrison, and A.J. Morton. 1986. Description and analysis of
vegetation. Pages 437-524 in P.D. Moore and S.B. Chapman, editors. Methods in plant
ecology. Blackwell Scientific Publications, Oxford, England.
Krebs, C.J. 1999. Ecological methodology. Addison-Wesley Educational Publishers, Inc.,
Menlo Park, CA. 620 pp.
Pringle, J.D. 1984. Efficiency estimates for various quadrat sizes used in benthic sampling.
Canadian Journal of Fisheries and Aquatic Sciences 41:1485-1489.
Wiegert, R.G. 1962. The selection of an optimum quadrat size for sampling the standing crop
of grasses and forbs. Ecology 43:125-129.
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