Principles of Plant Genetics and Breeding

andomization. The blocks should be laid across the
fertility gradient.
Probability
Statistical probability is a procedure for predicting the
outcome of events. Probabilities range from 0 (an event
is certain not to occur) to 1.0 (an event is certain to
occur). There are two basic laws of probability – product
and sum laws. The probability of two or more outcomes
occurring simultaneously is equal to the product
of their individual probabilities. Two events are said to
be independent of one another if the outcome of each
one does not affect the outcome of the other. Genetic
ratios may be expressed as probabilities. Consider a
heterozygous plant (Rr). The probability that a gamete
will carry the R allele is one-half. In a cross, Rr × Rr
(selfing), the probability of a homozygous recessive
(rr offspring) is 1 / 2 × 1 / 2 = 1 / 4 . The probability that
one or another of several mutually exclusive outcomes
will occur is the sum of their individual probabilities.
Using the cross Rr × Rr, the F 2 will produce
RR : Rr : rr in the ratio 1 / 4 : 1 / 2 : 1 / 4 . The probability
that a progeny will exhibit a dominant phenotype
(RR, Rr) = 1 / 4 + 1 / 2 = 3 / 4 . Other examples were discussed
in Chapter 3.
In using probabilities for prediction, it is important
to note that a large population size is needed for accurate
prediction. For example, in a dihybrid cross, the
F 2 progeny will have a 9 :3:3:1 phenotypic ratio,
indicating 9/16 will have the dominant phenotype.
However, in a sample of exactly 16 plants, it is unlikely
that exactly nine plants will have the dominant phenotype.
A larger sample is needed.
Measures of central tendency
The distribution of a set of phenotypic values tends
to cluster around a central value. The most common
measure of this clustering is the arithmetic mean. Plant
breeders use this statistical procedure very frequently in
their work. The formulae for calculating means are:
Sample mean, X ¯ =∑X/n
Population mean, µ=∑X/N
where X = measured value of the item, X ¯ = sample
mean, µ =population mean, n = sample size, and N =
population sample.
COMMON STATISTICAL METHODS IN PLANT BREEDING 149
Table 9.1 Ungrouped data for distribution of plant
seedling height.
or:
Distribution of seedling height (cm)
5 6 7 8 9 10 11 12 13 14 15 16 17 Total
F 1 5 14 16 12 3 50
F 2 4 10 13 17 20 28 25 18 17 13 11 10 7 193
Table 9.2 Grouped data for frequency calculation.
f 1 x i f i f i x i f i x i 2 f i x i f i x i 2
5 4 20 100
6 10 60 360
7 13 91 637
5 8 17 136 1,088 40 320
14 9 20 180 1,620 126 1,134
16 10 28 280 2,800 160 1,600
12 11 25 275 3,029 132 1,452
3 12 18 216 2,592 36 432
13 17 221 2,873 494 4,938
14 13 182 2,548
15 11 165 2,475
16 10 160 2,560
17 7 119 2,023
The sample mean is calculated as:
X¯ = X / n (for ungrouped data; Table 9.1)
X ¯ =∑X i f i /n (for grouped data; Table 9.2)
where X i = value of the ith unit included in the sample,
f i = frequency of the ith class, and n =∑f i .
The sample mean of seed size of navy beans is:
X ¯ =
n
∑
i=
1
n
∑
i=
1
i
X / n
i
n ∑f i x i ∑f i x i 2
193 2,105 24,705
= (17.2 + 18.1 + ...+ 19.7)/10
= 190.9/10
= 19.01 g per 100 seed