Principles of Plant Genetics and Breeding

Coefficient of variation
The coefficient of variation is a measure of the relative
variability of given populations. Variance estimates have
units attached to them. Consequently, it is not possible
to compare population measurements of different units
(i.e., comparing apples with oranges). For example, one
population may be measured in kilograms (e.g., yield),
while another is measured in centimeters or feet (e.g.,
plant height). A common application of variance is the
test to find out if one biological sample is more variable
for one trait than for another (e.g., is plant height in soybean
more variable than the number of pods per plant?).
Larger organisms usually vary more than smaller ones.
Similarly, traits with larger means tend to vary more
than those with smaller means. For example, grain yield
per hectare of a cultivar (in kg/ha) will vary more than
its 100 seed weight (in grams). For these and other
enquiries, the coefficient of variation facilitates the comparison
because it is unit free.
The coefficient of variation (CV) is calculated as:
CV = (s/X ¯ ) × 100
For the number of leaves per plant example:
CV = 1.37/7.9 = 0.173
= 17.3%
A CV of 10% or less is generally desirable in biological
experiments.
Standard error of the mean
The standard error measures the amount of variability
among individual units in a population. If several samples
are taken from one population, the individuals will
vary within samples as well as among samples. The standard
error of the mean (SE) measures the variability
among different sample means taken from a population.
It is computed as:
s x = s/√n
For the number of leaves per plant example:
s x = 1.37/√10
= 0.433
The standard error of the mean indicates how precisely
the population parameter has been estimated. It
COMMON STATISTICAL METHODS IN PLANT BREEDING 151
may be attached to the mean in the presentation of
results in a publication (e.g., for the leaves per plant
example, it will be 7.9 ± 0.43).
Simple linear correlation
Plant breeders are not only interested in variability as
regards a single characteristic of a population, but often
they are interested in how multiple characteristics of
the units of a population associate. If there is no association,
covariance will be zero or close to zero. The
magnitude of covariance is often related to the size of
the variables themselves, and also depends on the scale
of measurement.
The simple linear correlation measures the linear
relationship between two variables. It measures a joint
property of two variables. Plant breeding is facilitated
when desirable genes are strongly associated on the
chromosome. The relationship of interest in correlation
is not based on cause and effect. The degree (closeness
or strength) of linear association between variables is
measured by the correlation coefficient. The correlation
coefficient is free of scale and measurement, and has
values that lie between +1 and −1 (i.e., correlation can
be positive or negative). If there is no linear association
between variables, the correlation is zero. However, a
lack of significant linear correlation does not mean there
is no association (the association could be non-linear or
curvilinear).
The population correlation coefficient (ρ) is given by:
ρ=σ 2 XY /√(σ X 2 ×σ Y 2 )
where σ X 2 = variance of X, σY 2 = variance of Y, and σ 2 XY =
covariance of X and Y. The sample covariance is
called the Pearson correlation coefficient (r) and is
calculated as:
r = s 2 XY /√(s 2 X × s 2 Y )
where s 2 XY = sample covariance of X and Y, s 2 X = sample
variance of X, and s 2 Y = sample variance of Y.
The computational formula is:
r = [N(∑XY ) − (∑X)(∑Y )]/
√[N(∑X 2 ) − (∑X) 2 ][N(∑Y 2 ) − (∑Y ) 2 ]
The data in Table 9.3 shows the seed oil and protein
content of 10 soybean cultivars. The calculation yields
the following results: