9597

• UCREFRP Catalog Number: 9597
• Author: U.S. Environmental Protection Agency.
• Title: Biological Field and Laboratory Methods for Measuring the Quality of Surface Waters and Effluents.
• USFW Year: 1973.
• USFW - Doc Type: Cincinnati, OH.
• Copyright Material: NO

BIOLOGICAL METHODS

Standard deviation of the mean or standard
error (s- ):
x

-1:2 S
Sx: -V sj{ = -.fii

(15)

4.2 Statistics for Stratified Random Samples

The calculations of the sample statistics for
stratified random sampling are as follows (see
2.2.2 Stratified random samples):
For the mean of stratum k:

nk
~ Yki
_ i=1
y=-
nk

(16)

Le., simply compute an arithmetic average for
the measurements of stratum k.
For the variance of stratum k:

S2 =

nk ( nk ~
~ Yki2 - ~ Yki 2
i=1 i=1
nk

(17)

nk -1

i.e., simply Equation 12 applied to the data of
the kt h stratum.
For the mean of the stratified sample:

m
~ NkYk
_ k=1
Yst = N

(18)

for either type allocation or alternatively for
proportional allocation:

m
~ nkYk
- k=1
Yst =

(19)

n

No te that Equations (18) and (19)
identical only for proportional allocation.

4.3 Statistics for Subsamples

If simple random sampling is used to select a
subsample, the following formulas are used to
calculate the sample statistics (see 2.3 Sub-

sam piing) :
For the sample mean:

t

= 1 ~ (LiYi)
y= - .... ---:-
n i=1 Zl
n~Li
i=1

(20)

where y is the average, computed over sub-
samples as well as for the sample
14
_ j~1 Yi j (21)
Yi =--n-

where Yi j equals the observation for the {.!!
element in the itl! primary unit, and Lj is the
number of observations upon elements for
primary unit L
For the variance of the sample mean:

n 1\ ^
s2 (y) = ~ (Yi - Yn)2
n i-I
n(n-l) ( ~ lj)2 -
i=1

(22)

1\
where Yi is computed as

~. = LiYi
1 Zi

(23)

4

^
where Yn is computed as

! 1 n ^ = n
Yn =- ~ Yi = Y ~ lj
n i=1 i=1

(24)

or alternatively

A
= 1 ^ (~ Yj}2
s2 (y) = . ~i-- .
n n
n(n-1) (~ Lj}2
i=1

(25)

are

4.4 Rounding

The questions of rounding and the number of
digits to carry through the calculations always
arise in making statistical computations.
Measurement data are approximations, since
they are rounded when the measurements were
taken; count data and binomial data are not
subject to this type of approximation.
Observe the following rules when working
with measurement or continuous data.

. When rounding numbers to some number
of decimal places, first look at the digit to the

~

10