An Introduction to Determining Optimum Quadrat Size and Shape

Problem Set #2: An Introduction to Determining
Optimum Quadrat Size and Shape
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 kangaroos or caribou 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? The answer to these questions is far from simple.
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, 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 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 statistical
precision as the lowest standard error of the mean and the arrowest 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 by keen biologists who 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. This positive bias 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 1 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 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 data set you are working on, you should immediately disqualify the smallest
quadrat size from consideration to avoid bias from the edge effect.
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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). Areas are never 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 and 4m x
0.25 m. 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 on its own. 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 other hand, an
equal number of quadrats were to be taken for each quadrat size, one would prefer the long, thin
quadrat shape.
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Table 1. Effect of quadrat size on SD for measurements of basal area of trees in an oak-hickory
forest in North Carolina. The data are from Bormann (1953).
Quadrat size Observed SD
(m) per 4 m 2
Sample size a
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
a 2
The number of quadrats of a given size needed to sample 1,120 m .
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 m 2 . This conversion is simple for means, SDs, and SEs: divide by the relative area.
For example,
Mean number per m 2 2
Mean number per 0.25m
=
0.25
SD per m 2 2
SD per 4m
=
4
For variances, the square of the conversion factor is used:
Variance per m 2 =
2
Varianceper
9 m
2
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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.
Wiegert’s method
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,
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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:
Quadrat size (area)
1 3 4 12 16
Observed variance per 0.25 m 2 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).
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.
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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 2
Sample sizes are approximately the number to make a total sampling area of 20 m .
b 2
All expressed per m .
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
Relative cost =
Time to take1sampleof
a givensize
Minimum time to take1sample
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:
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Equation 1

Relative variance =
(SD)
2
(MinimumSD)
2
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Equation 2
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, but this was not tested for by Pringle. In case you are wondering, the 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 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 Glacier National Park, Montana
and divided into 16 quadrats of 1-m 2 . The numbers of queencup, Clintonia uniflora, 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.
Use the 3 quadrats provided on the overhead transparency. 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 randomly-located point.
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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 take
1 sample
(min)
10m x 10m 178 3
25m x 25m 28 8
50m x 50m 7 15
a
Sample sizes are approximately the number of quadrats to make a total sampling area of
17,860m 2 .
b 2
All expressed per m .
Table 4. Wiegert’s method to determine optimal quadrat size for counting balsam fir in a
northern hardwood stand.
Quadrat size Relative Relative Product of relative
(m) variance cost variance and
relative cost
10m x 10m
25m x 25m
50m x 50m
SOME MORE TO THINK ABOUT
1. What is the most optimum quadrat size to count balsam fir in this northern hardwood stand?
2. 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.
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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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