Agriculture Reference
In-Depth Information
1000
800
600
95%
400
2
σ
2.5%
2.5%
200
0
500
510
520
530
540
550
560
570
580
590
600
610
620
µ
Figure 5.2
95% of a population which is normally distributed will lie within one standard deviation from the population
mean.
equations for
s
2
2
:
σ
sigma) and is defined thus:
and
n
n
n
2
σ
=
√
n
n
2
1
(
x
i
−
µ)
/
s
2
x
i
)
−
ˆ
=
1
(
1
(
x
i
)
(
n
−
1
)
i
=
i
=
i
=
n
2
ˆ
and the latter is represented by the symbol
s
, where:
n
n
n
n
2
x
i
)
−
σ
=
1
(
1
(
x
i
)
=
√
2
ˆ
1
(
x
i
−¯
)
/(
−
)
s
x
n
1
i
=
i
=
i
=
Although these
look
more complicated than those
given previously, they are
easier to use
because the mean
does not have to be worked out first (which would entail
entering all the data into the calculator twice). Note the
Another measure of the spread of data around the
mean
is the
variance
, which is the square of the standard
deviation. The
estimated variance
of a population is
given by:
difference between
(
x
i
)
(each value of
x
squared and
then the squares totalled) and [
(
n
2
2
x
i
)
]
(the values of
x
totalled and then the sum squared).
Standard deviations, as measures of spread around
the mean, are probably intuitively more understandable
than variances, for example, 68% of the population
fall within one standard deviation of the mean. Why
introduce the complication of variances? Well,
variances
are additive
in a way standard deviations are not. Thus,
if the variances attributable to a variety of factors have
been estimated, it is mathematically valid to sum them
to estimate the variance due to all the factors acting
together. Similarly, a total
variance
can be
partitioned
into the variances attributable to a variety of individual
factors. These operations, which are used extensively in
quantitative genetics, cannot so readily be performed
with standard deviations.
s
2
ˆ
=
1
(
x
i
−¯
)
(
−
)
x
n
1
i
=
and the
actual variance
of a sample drawn from a pop-
ulation (or of an entire population if every member of
it has been measured) is given by:
n
2
n
2
σ
=
1
(
x
i
−
µ)
=
i
Calculators are often programmed to give means
and either standard deviations or variances with a few
key strokes once data have been entered. However, in
case it is necessary to derive these descriptive statistics
semi-manually, it is useful to know about alternative
