Principles of Plant Genetics and Breeding

150 CHAPTER 9
Using data in Table 9.2, the mean of the F 2 can be
obtained as:
X ¯ =∑X i f i /n
= 2,105/193
= 10.91 cm
Measures of dispersion
Measures of dispersion or variability concerns the
degree to which values of a data set differ from their
computed mean. The most commonly used measure of
dispersion is the mean square deviation or variance. The
population variance is given by:
σ 2 = [∑(X 1 −µ) 2 ]/N
where σ 2 = population variance, X 1 = value of observations
in the population, µ=mean of the population, and
N = total number of observations in the population.
The sample variance is given by:
s 2 = [∑(X − X ¯ ) 2 ]/(n − 1)
where s 2 = sample variance, X = value of the observation
in the sample, X ¯ = mean of the sample, and n = total
number of observations in the sample.
The computational formula is:
s 2 = [∑X 2 − (∑X ¯ ) 2 /n]/(n − 1)
Using the data below for number of leaves per plant:
Number of leaves Total
X 7 6 7 8 10 7 9 8 7 10 79 =∑X
X 2 49 36 49 64 100 49 81 64 49 100 641 =∑X 2
(∑X) 2 /n = 79 2 /10 = 6,241/10 = 624.1
s 2 = (641 − 642.1)/9 = 16.9/9 = 1.88
Variance may also be calculated from grouped data.
Using the data in Table 9.2, variance may be calculated
as follows:
s 2 = [n∑fx 2 − (∑fx) 2 ]/n(n − 1)
= [193(24,705) − (2,105) 2 ]/193(193 − 1)
= (4,768,065 − 4,431,025)/37,056
= 337,040/37,056
= 9.10 (for F 2 , the most variable generation
following a cross)
Variance for the F 1 is 1.67.
Standard deviation
The standard deviation (SD) measures the variability
that indicates by how much the value in a distribution
typically deviates from the mean. It is the positive square
root of the population variance. The larger the value of
the standard deviation, the more the observations (data)
are spread about the mean, and vice versa.
The standard deviation of the sample is simply:
s =√s 2
For the number of leaves per plant example:
s =√1.88
= 1.37
Similarly, for the seedling height data:
SD of the F 2 =√9.1 = 3.02
SD of the F 1 =√1.67 = 1.29
Normal distribution
One of the most important examples of continuous
probability distribution is the normal distribution or
the normal curve. It is important because it approximates
many kinds of natural phenomena. If the population
is normally distributed, the mean = 0.0 and the
variance = 1.0. Further, a range of ±1 SD from the
mean will include 68.26% of the observations, whereas
a range of ±2 SD from the mean will capture most of
the observations (95.45%) (Figure 9.1). The shape
of the curve varies depending on the nature of the
population.
68.3%
95.4%
99.7%
–3σ –2σ –1σ µ +1σ +2σ +3σ
Figure 9.1 Normal distribution curve.