Stratified SamplingStratified Sampling

Lecturer: Chad JensenLecturer: Chad Jensen

Sampling MethodsSampling Methods

SRS (simple random sample)SRS (simple random sample) SystematicSystematic ConvenienceConvenience JudgmentJudgment QuotaQuota SnowballSnowball Stratified SamplingStratified Sampling

What is Stratified Sampling?What is Stratified Sampling?

StratificationStratification is the process of is the process of grouping members of the population grouping members of the population into relatively homogeneous into relatively homogeneous subgroups before sampling. subgroups before sampling.

AdvantagesAdvantages

Provides greater precision than a SRS Provides greater precision than a SRS (simple random sample) of the same (simple random sample) of the same sizesize

Often requires a smaller sample, Often requires a smaller sample, which saves money which saves money

Can guard against an Can guard against an "unrepresentative" sample"unrepresentative" sample

Focuses on important subpopulations Focuses on important subpopulations but ignores irrelevant onesbut ignores irrelevant ones

DisadvantagesDisadvantages

Can be difficult to select relevant Can be difficult to select relevant stratification variablesstratification variables

Often requires more administrative Often requires more administrative work than an SRSwork than an SRS

Not useful when there are no Not useful when there are no homogeneous subgroupshomogeneous subgroups

Can be expensiveCan be expensive

Proportionate StratificationProportionate Stratification

Each Stratum has the same sampling Each Stratum has the same sampling fraction.fraction.– Can provide better precision than a SRS Can provide better precision than a SRS

of the same size. of the same size. – Gains in precision are greatest when Gains in precision are greatest when

values within strata are homogeneous.values within strata are homogeneous.– Gains in precision accrue to all survey Gains in precision accrue to all survey

measures. measures.

Proportionate StratumProportionate Stratum

nnhh = ( N = ( Nhh / N ) * n / N ) * n

nnh h = is the sample size for stratum h.= is the sample size for stratum h.

NNh h = is the population size of stratum = is the population size of stratum h.h.

N = the total population sizeN = the total population size n = the total sample sizen = the total sample size

Disproportionate Disproportionate StratificationStratification

The sampling fraction may vary from one The sampling fraction may vary from one stratum to the next. stratum to the next. – If variances differ across strata, If variances differ across strata,

disproportionate stratification can provide disproportionate stratification can provide better precision than proportionate better precision than proportionate stratification, when sample points are correctly stratification, when sample points are correctly allocated to strata. allocated to strata.

– The researcher can maximize precision for a The researcher can maximize precision for a single important survey measure. single important survey measure.

– Gains in precision may not accrue to other Gains in precision may not accrue to other survey measures. survey measures.

Disproportionate StratumDisproportionate Stratum

nnhh = n * ( N = n * ( Nhh * S * Shh ) / [ Σ ( N ) / [ Σ ( Nii * S * Si i ) ] ) ]

nnh h = sample size for stratum h.= sample size for stratum h. n = total sample sizen = total sample size NNh h = population size of stratum h.= population size of stratum h.

SSh h = Standard deviation of stratum h= Standard deviation of stratum h

Proportionate vs. Proportionate vs. DisproportionateDisproportionate

Disproportionate can be a better Disproportionate can be a better choice (e.g., less cost, more choice (e.g., less cost, more precision) if sample elements are precision) if sample elements are assigned correctly to strata. assigned correctly to strata. – Example: Given a fixed budget or fixed Example: Given a fixed budget or fixed

sample size, how should sample be sample size, how should sample be allocated to get the most precision from allocated to get the most precision from a stratified sample? a stratified sample?

Proportionate vs. Proportionate vs. DisproportionateDisproportionate

Recommendation:Recommendation:

If costs and variances are about If costs and variances are about equal across strata, choose equal across strata, choose proportionate stratification. proportionate stratification.

If they differ, consider If they differ, consider disproportionate stratification. disproportionate stratification.

ExampleExample

StratumStratum Mean ScoreMean Score Standard Standard DeviationDeviation

BoysBoys 7070 10.2710.27

GirlsGirls 8080 6.666.66 The state administers a reading test The state administers a reading test

to a sample of to a sample of 3636 third graders. third graders. The school system has The school system has 20,00020,000 third third

gradersgraders 10,00010,000 boys and boys and 10,00010,000 girls. girls.

Proportionate StratumProportionate Stratum

nnhh = ( N = ( Nhh / N ) * n / N ) * n

18 boys = (10,000/20,000) *3618 boys = (10,000/20,000) *36 18 girls = (10,000/20,000) *3618 girls = (10,000/20,000) *36

Disproportionate StratumDisproportionate Stratum

StratumStratum Mean ScoreMean Score Standard Standard DeviationDeviation

BoysBoys 7070 10.2710.27

GirlsGirls 8080 6.666.66 nnhh = n * ( N = n * ( Nhh * S * Shh ) / [ Σ ( N ) / [ Σ ( Nii * S * Si i ) ] ) ]

21.83 boys21.83 boys = 36 * ( 10,000 * 10.27 ) / = 36 * ( 10,000 * 10.27 ) / [ ( 10,000 * 10.27 ) + ( 10,000 * 6.67 ) ][ ( 10,000 * 10.27 ) + ( 10,000 * 6.67 ) ]

14 girls14 girls = (36 – 22 boys) = (36 – 22 boys)

ConclusionConclusion

How can you use stratified sampling How can you use stratified sampling in your project?in your project?

Questions? Comments? Questions? Comments? Concerns? Emotional Concerns? Emotional

Outburst?Outburst?