Stratified Sampling

Basic idea of stratified sampling

population is divided into non-overlapping groups/subpopulations called strata

why stratified sampling

improved accuracy

cost of obtaining observations may be less

separate estimates may be desired for individual strata

L

number of strata

Ni

population size of stratum

N

overall population size N1 + … + NL

ni

sample size for each stratum

n

overall sample size

proportion allocation

sample sizes for each stratum are chosen proportionally to the population sizes

• ensures that each stratum is represented exactly in proportion to its size in the population

• overall sample mean is an unbiased estimator of the population mean

• minimizes variance

does the population mean change when you change allocation

no

when is non-proportion allocation preferred

When one strata has much higher variability this results in a reduction of the overall variance of the estimate.

allocation scheme plays an important role in accuracy of estimates (T or F)

True

Is stratified sampling always better than simple random sampling?

With proportion allocation stratified sampling gives better estimates (for pop mean) than SRS. But it is not true generally

when does stratified give better estimates than SRS

when there are large differences between strata in their means.

another time when stratified sampling causes high variance

when some stratums variances are much higher than others. (under proportion allocation)

why not use overall sample mean to estimate the population mean?

Stratified mean corrects the weights so each startums contribution is proportional to its actual population size.

overall mean doesn’t account for stratified differences (leads to higher variance)

we want to also avoid overrepresentation or underrepresentation

optimal allocation considerations

• larger sample size for stratum with larger population size

• larger sample size for stratum with larger variance

• Larger sample size for stratum with lower cost

what ni do we want to chose?

The ni that minimizes variance V for a fixed cost C

fixed error for choosing n

we calculate the minimum necessary sample size n to keep the variance within the bound.

choosing n for a fixed cost C

allocate sample sizes across strata to minimizes variance within budget constraints.

When is post stratification nearly as accurate as startified random sampling

if Ni/N is known and ni >= 20

when is post stratification feasible

when we can’t stratify sampling units until after a sample has been selected (respondents in a telephone poll gender classification)

difference between stratum sizes in post stratification

they are random and may deviate considerably from Ni/N

post stratification error bound is smaller than simple estimation with SRS

True , simple sample mean gives an underestimation compared with ypost.