Environmental Sampling Design - Fundamentals of Environmental Sampling and Analysis - page 55

by the following formulas:

x ¼ X

r

w k x k ð3 : 2Þ

k¼1

¼ X

r

w k s k
n k

s 2

ð3 : 3Þ

k¼1

where w k ¼ the fraction or the weight of the population represented by stratum k.
Eqs 3.2 and 3.3 indicate that the mean and standard deviation for stratified random
sampling are the weighted average of all strata. The weight, w k , is assumed to be
known before any sampling takes place.
The following discussions focus on various methods to allocate the number of
samples into each stratum. Due to the complex nature of the stratified random
sampling, interested readers are referred to Gilbert (1987) for a more thorough
discussion on these mathematical equations.
Equal allocation: Each stratum is assigned the same number of samples. Since
the total number of stratum is r, the number of sample in each stratum ðn k Þ¼n

= r,

where n is the total number of samples from all strata.
Proportional allocation: The number of samples in each stratum is proportional
to the size of the stratum. The larger the stratum, the more the sample can be
collected. Assume N¼ total population units, N k ¼ total population units in stratum
k, n¼ total sample units, and n k ¼ total sample units in stratum k, then:

n k
n ¼

N k
N w k

or n k ¼ nw k ð3 : 4Þ

Optimal allocation: The cost is considered through either the optimal precision for a
fixed study cost or the optimal cost for a fixed level of precision. The number of
samples in the kth stratum is related to the variation (s k ) and the cost per sampling
unit in the kth stratum (C k ) by the following equation:

w k s k =

p

C k

n k ¼ n ð3:5Þ

X

r
k¼1 ðw k s k =

C p Þ

A special case is when the sampling cost per sampling unit is the same for all strata,
then the above equation reduces to Eq. 3.6, which is frequently called Neyman
allocation.

w k s k
X

n k ¼ n ð3 : 6Þ

r

w k s k

k¼1