Agriculture Reference
In-Depth Information
values of A to the variability in A from experiment to
experiment, and also a statistical table in which the
probability of obtaining such a difference by chance,
given the number of plants measured, can be looked
up. The equation is:
mean. That is:
variance of
n
B
1
plants
n
As an example if this consider that the height of 50
individual plants (i.e.
n
2
σ
B
1
=
=
50) of a pure-line barley cul-
tivar was recorded and that the average plant height of
all plants measured was calculated to be 100 cm, with a
variance of 40 cm
2
. The variance and the standard error
of the mean of this sample of plant heights would be:
Actual value ofA-predicted value of A
standard error of A
t
=
A
−
0
t
=
se
(
A
)
A
variance of 50 plants
50
=
t
=
Variance of mean
(
)
se
A
25
50
=
In fact, as I am sure that you have all noticed, the A-
scaling test is just a particular application of Student's
t
test, which you may have come across elsewhere.
In order to calculate
t
, A has to be divided by its
standard error
(se). It is known that A
0.80 cm
2
=
=
σ
=
√
(
And the standard error
0.80
)
=
0.89 cm
±
Therefore the mean is 100 cm
0.89 cm.
2 B
1
−
F
1
−
P
1
, where B
1
, F
1
and P
1
, are the measured (not the
predicted
) means of the B
1
,F
1
and P
1
generations. But
what is the standard error of A? A standard error is, like
a standard deviation, the square root of a variance as
shown earlier. In fact, the standard error of A is the
square root of the variance of the mean of A (
=
So, given a set of individual measurements, you
should be able to calculate the mean, the variance and
the standard deviation of the population of which the
data you are given can be assumed to be an unbiased
sample. You should also be able to calculate both the
variance of the mean and thus its standard error.
Now consider a simple example where two homozy-
gous barley cultivars (P
1
and P
2
) are cross pollinated
and a sample of F
1
seed is backcrossed to the higher
yielding parent (P
1
) to produce B
1
seed. Now if 11
P
1
,11F
1
and 11 B
1
seeds were planted in a properly
randomized experiment and the height of each plant
recorded and the following means and standard errors
are determined.
P
1
=
2
σ
A
), and
σ
A
. Therefore:
is represented as
2
2
2
2
P
1
σ
A
=
σ
B
1
+
σ
F
1
+
σ
4
2
B
1
where
σ
is the variance of the mean of the B
1
genera-
2
σ
tion,
F
1
is the variance of the mean of the F
1
generation
2
and
P
1
is the variance of the mean of the P
1
generation.
It is essential to realize that the variance of the mean
of a generation (i.e.
σ
F
1
=
±
±
109.1
9.1 :
105.5
8.6 :
2
is not the same as the vari-
ance between plants in that generation. In principle,
the former is calculated by growing adequate numbers
of plants representing the generation in several differ-
ent experiments, calculating a generation mean for each
experiment and then calculating the variance of these
different means (effectively treating them as raw data).
A variance of the mean so determined is less than the
variance between all the plants grown in all the experi-
ments. Fortunately, it is not necessary to perform several
different experiments as described. Statisticians have
demonstrated that a satisfactory estimate of the vari-
ance of the mean is obtained by dividing the variance
derived from a single sample of plants by the number of
plants measured that contribute to the estimate of the
σ
B
1
)
B
1
=
±
108.3
10.0
Now the variance of each family would be:
2
2
2
2
2
2
σ
P
1
=
(
9.1
)
:
σ
F
1
=
(
8.6
)
:
σ
B
1
=
(
10.0
)
2
Therefore it follows that the variance of A (
σ
A
)
would be:
2
2
2
2
P
1
σ
A
=
σ
B
1
+
σ
F
1
+
σ
4
2
2
2
=
(
)
+
(
)
+
(
)
=
4
10.0
8.6
9.1
200
and the standard error of A (
σ
A
) is given by:
σ
A
=
√
(σ
)
=
√
(
2
A
200
)
=
14.1
