6.3: Distance Methods

Video Presentation

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Learning Guide


Density is the number of individuals per unit area, which is inseparably linked to the closeness of individuals to one another. Consider an elevator that is 10′ x 10′. When it begins its ascent there are 2 people on the elevator and they are able to maintain a reasonable amount of “personal space” (Figure 1a). As the elevator ascends, it stops at other floors to admit more passengers, who must stand closer and closer together in the limited space. Finally, the occupants are actually rubbing shoulders as the elevator reaches the top floor (Figure 1b).


Figure 1. The relationship between density and spacing between individuals is illustrated by scenes from elevators: a) 2 people in the elevator have sufficient space to maintain greater distance between them compared to b) as the density of individuals in a defined space increases, the distance between individual and space that each person occupies decreases.


The same concept is true for plants. The more plants that are rooted in a fixed area, the closer those plants must be to one another. Consequently in low-density areas, distances between plants are large (Figure 2a) compared to high-density areas, where the distances between plants are smaller (Figure 2b).


Figure 2. The relationship between plant density and distances between plants (blue dashed lines): a) in low-density areas, plants tend to be spaced at greater distances; b) in areas with higher plant density, the spacing or distances between individuals is decreased.


Distance methods are a family of measurement approaches that are an alternative to plot-based, counting methods.

Mean Area

Distance methods are based on the concept of mean area.

  • The area occupied by an individual plant is related to the spacing between plants, not to the plant’s physical size.  Therefore, we focus on the distance between plants, which can be conceptualized as a radius of an imaginary circle around a plant (Figure 3a).
  • As plant density increases, the radius of the imaginary circle decreases, and the area that each plant occupies decreases (Figure 3b).


Figure 3. When only two plants are present in a plot, they have a particular area they occupy that is related to the distance between their stems (a); however, when more plants are added, the distance between plant stems decreases and the area occupied by each plant decreases (b).

  • By measuring distances between plants, we can calculate the mean area (MA) per plant.
  • Mean area is inversely related to plant density. Both describe the relationship of individual plants on an area basis:

So, D = 1/MA   and MA = 1/D

Advantages of Distance Methods

  • Eliminates edge-effects and boundary decisions: Distance methods do not use quadrats or plots. Observers do not have to make decisions when plant bases are adjacent to or intersected by the edge of the plot. Since there is no need to decide whether a plant is “in” or “out”, this reduces the amount of time spent on boundary decisions and the potential for non-sampling errors and associated bias.
  • Efficient in forests, woodlands, and some shrublands: Measuring the density of large individuals in plots can be difficult and inefficient when plants are closely spaced or we want to measure density over large areas. In woodland and forests, it can be difficult to establish and keep track of the plot boundaries. Distance methods eliminate this problem because the focus is on distance between plants, and plots are not used.

Disadvantages of Distance Methods

  • Assume random distribution of plants: Most distance methods assume that plants are randomly distributed throughout the study area.  Most plants are not randomly distributed, and so caution should be applied when interpreting the results from distance methods.  The Wandering Quarter method does not assume a random distribution of plants.

Distance Methods

Most distance methods share the following characteristics:

  1. Distance measurements (d) are taken along transects in the study area.
  2. A mean distance (  ) is calculated for each transect by averaging the distance measurements along the transect.
  3. The mean distance (  ), is converted to mean area (MA). Conversion equations are specific to the distance method used.


The most common distance methods are:

  1. Nearest Individual
  2. Nearest Neighbor
  3. Random Pairs
  4. Point-Centered Quarter
  5. Wandering Quarter

Nearest Individual

The nearest individual method measures the distance (d) from a sampling point on a transect to the stem or base of the plant closest to the sampling point (Figure 4).


Figure 4. Conceptual illustration of measurements using the nearest individual method. The orange lines mark the distance (d) measured between the sampling point on the transect and the nearest individual plant.


The process is repeated at each sampling point along the transect. Interestingly, the actual distance between plants is never directly measured with this method.

Nearest Neighbor

Unlike the nearest individual method, the nearest neighbor method measures distances between plants. The individual nearest to the sampling point on the transect is identified; we refer to that individual as the “focal plant” (Figure 5).


Figure 5. Conceptual illustration of measurements using the nearest neighbor method. Blue lines mark the “focal plant” or individual nearest to the sampling point on the transect. The orange lines mark the distance (d) measured between the focal plant and its nearest neighbor.


Then, the individual plant that is closest to the focal plant is identified as the “nearest neighbor”, and the distance (d) between these two plants in measured.


Nearest neighbor has been used to assess the effects of intraspecific competition in forested ecosystems. Pielou (1960) studied populations of ponderosa pine and found a positive correlation between nearest neighbor distances and the sum of the trunk circumferences of the two neighboring plants. The trunk circumference was assumed to reflect the sizes of the root systems, suggesting that greater distances between trees allowed for larger root systems and greater tree trunk diameter.

Random Pairs

The random pairs method also requires identification of the “focal plant”, the nearest individual plant to the sampling point. An imaginary line is then drawn through the sampling point; this line is perpendicular to the line between the sampling point to the focal plant (Figure 6).


Figure 6. Conceptual illustration of measurements using the random pairs method. Blue lines mark shows the “focal plant” or nearest individual to the sampling point on the transect. The red lines are perpendicular to the blue lines and are drawn through the sampling point on the transect.  The orange lines mark the measured distance (d), between the focal plant and the closest individual that exists on the opposite side of the perpendicular line.


The observer then needs to examine the plants that are located on the opposite side of the perpendicular line and identify the plant that is closest to the focal plant. This is repeated for each sampling point along the transect.

Point-Centered Quarter

The point-centered quarter method measures 4 distances (d) for each sampling point. An imaginary line, perpendicular to the transect, is drawn through the sampling point: this delineates 4 quadrants (or quarters) similar to an x,y coordinate system in which the sampling point is at the origin (Figure 7).


Figure 7. Conceptual illustration of measurements using the point-centered quarter method. Blue lines mark the quarters delineated relative to the sampling points, and the orange lines mark the distance between the sampling points and the closet individual in each quarter. An average value ( p ) is calculated for each sampling point.


Then, in each quarter, the individual that is closest to the sampling point is identified and the distance (d) between this individual and the sampling point is measured. The four distances are then averaged to obtain a mean distance for each sampling point.  The transect is the sampling unit, so the mean distances from the sampling points are again averaged to obtain a total mean distance (  ) for the entire transect.

Calculating Mean Area and Density

Each method uses a specific conversion equation to convert the average distance (  ) for the transect to mean area. Remember that each transect is a sampling unit, so there should be one average distance (  ) value for each transect. Us the appropriate conversion equation to convert the average distance to MA.


Conversion Equations by Method

  • Nearest Individual:                                                   MA = (2  )2
  • Nearest Neighbor:                                                   MA = (1.67  )2
  • Random Pairs:                                                         MA = (0.87  )2
  • Point-Centered Quarter:_____________________MA = (  )2


Convert Mean Area (MA) to Density (D)

The conversion from mean area (MA) to density (D) is straightforward, because these two values (MA and D) are the inverse of each other:

D = 1/MA

Example Calculation:

We will use measurements taken using the random pairs method to calculate mean are and density (Figure 8).


Figure 8. Example of measurements taken using the random pairs method. Three measurements were taken: d1 = 4.0 m, d2 = 2.75 m, d3 = 3.25 m.


1. Calculate the average distance (  ) for the transect:

2. Calculate mean area (MA):

3. Calculate density (D):

Wandering Quarter Method

While the four methods described above are simple and objective, they are suitable only when plants are randomly distributed on the landscape, which is rarely the case. Plants are frequently clumped or patchy in their distribution because of soil gradients, variable topography or heterogeneous seed distribution. Therefore, the assumptions of these methods are rarely met and their application is limited.


The wandering quarter method was designed for use in plant populations with non-random, aggregated distributions. The wandering quarter method does not use a straight transect tape. The observer proceeds in a predetermined direction until the first target plant is reached; this target plant becomes the focal plant and the first sampling point (Figure 9).


Figure 9. Conceptual illustration of measurements using the wandering quarter method. Blue lines mark the 90° quarter at each focal plant and orange lines mark the distance (d) to the individual within the 90° quarter that is closest to the focal plant. The closest individual identified then becomes the next sampling point and focal plant for the next measurement.


A “quarter” with a 90° (right angle) is established and maintained at a constant angle, so that the original direction of the transect is maintained along the 45° position of the quarter, essentially bisecting the quarter. The observer locates the closest plant that occurs within the boundaries of the 90° quarter, and the distance (d) between the focal plant and the closest plant is measured.


The closest plant then becomes the next focal plant and sampling point, and the orientation of the quarter is maintained. Thus, as illustrated in Figure 9, the quarter “wanders” from focal plant to focal plant, until a pre-determined number of sampling points and distances have been measured. Hence the name “wandering quarter” is applied to this method.


Conversion of distances to mean area and density are more complicated than for the previously discussed methods, and will not be presented here. Readers are referred to Catana (1963) and related papers for details about density calculations for this method.


Distance methods are rarely applied in rangeland monitoring on grasslands. However, they are used with some regularity in woodland and forest ecosystems. In South Africa, researchers used the wandering quarter method to determine shrub density. Shrub density estimates were then correlated to shrub biomass production per unit area and used to develop a fire behavior model for that ecosystem (Van Wilgen et al 1985).  It is important to be aware of these methods and to understand their strengths and weaknesses as alternatives to counting methods.

Self-check Activity

1. Distance methods are not the most common measurement method in rangeland management and tend to be more common in ecology. However, what are some advantages to using distance methods for measuring density? Select all that apply.

2. What is a disadvantage to using distance methods for measuring density?