Environmental Engineering Reference
In-Depth Information
Box 2.1
Parametric and non-parametric measures of precision.
By
parametric
, we mean an approach that assumes an underlying probability
distribution such as the Normal (Gaussian or bell-shaped) distribution. There
are several related parametric measures of precision. The
sampling variance
of
parameter
,
denoted VAR( ), can be estimated in many ways, depending on
the sampling process. In the special case where
is a population mean, the
sampling variance is a function of the
population variance
,
s
2
, and sample
size,
k
:
ˆ
ˆ
ˆ
s
2
(
ˆ
)
k
VAR(
ˆ
)
2
) is the standard function avail-
able on calculators and spreadsheets, that is, given
k
individual observations,
x
i
:
Population variance (often alternatively denoted
s
2
(
ˆ
)
(
x
i
ˆ
)
2
k
1
The square root of the population variance gives the
standard deviation
, while
the square root of the sampling variance gives the
standard error
:
SE(
ˆ
)
VAR(
ˆ
)
The
coefficient of variation
(CV) is the standard error expressed as a proportion
(or percentage) of the parameter estimate:
SE(
ˆ
)
ˆ
CV(
ˆ
)
The parameters discussed in this chapter are generally not normally distributed,
however, when sample size is reasonably large, a normal approximation for the
confidence interval
(CI) is often reasonable:
CI (
ˆ
)
ˆ
t
2,
SE(
ˆ
)
where value
t
2,
is taken from the two-tailed
t
-distribution. For a 95% confidence
interval,
0.05, and for large sample sizes,
t
approaches 1.96. When precision
is relatively low, it may be more appropriate to use a log-normal approximation
for the confidence interval, giving intervals that are asymmetric and constrained
to be positive:
CI (
ˆ
)
(
ˆ
/
w
,
ˆ
w
)
CV(
ˆ
)
2
)
w
exp
t
2,
ln(1
