Stratified Random Sampling
When we know that parts of the population differ with respect to the
quantity that we are estimating, we can obtain better estimates by using
stratification. A stratified random sample is obtained by separating the
population elements into nonoverlapping groups, called strata, and then se-
lecting a simple random sample from each stratum. Stratification is particu-
larly useful because: i) it can yield smaller error bounds than SRS, especially
when the measurement is homogeneous within strata, ii) The cost per obser-
vation can be lowered by appropriate choice of strata, and iii) it will yield
stratum-specific estimates of population parameters.
Drawing a Stratified Random Sample, notation: Choose strata,
then take a SRS from each. L= number of strata, Ni= population size
from stratum i,N=N1+N2+... +NL= total population size, ni= sample
size from stratum i, n =n1+n2+... +nL= total sample size.
Estimation of a population mean and total from stratified ran-
dom sampling: To estimate the mean µfrom stratified random sampling,
we use the sample stratified mean:
bµ= ¯yst =1
N[N1¯y1+N2¯y2+... +NL¯yL] = 1
N
L
X
i=1
Ni¯yi.
Since the strata are independent, we obtain the estimated variance by
summing the estimated variances from each stratum:
ˆ
V(¯yst) = 1
N2
L
X
i=1
N2
iµNi−ni
Ni¶s2
i
ni
To estimate the total τfrom stratified random sampling, we just estimate
the total τifrom each stratum and sum. Also, since the strata are indepen-
dent, the estimated variance of the total is just the sum of the estimated
variances from each stratum:
bτ=
L
X
i=1
Ni¯yiand
ˆ
V(bτ) =
L
X
i=1
N2
iµNi−ni
Ni¶s2
i
ni
1