2 & 3 STAGE Sampling

2 & 3 STAGE SAMPLING : 

2 & 3 STAGE SAMPLING 1

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2-Stage sampling is named so, because the sample is taken in two steps- The first is to select a sample of units, often called the primary units. Second is to select a sample of second stage units or subunits from each chosen primary unit. TWO-STAGE SAMPLING 2

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POPULATION THESE ARE THE PRIMARY UNITS THESE ARE THE SECONDARY UNITS N=81, n=5, M=9, m=3 SCHEMATIC REPRESENTATION OF 2-STAGE SAMPLING 3

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WHY 2-STAGE SAMPLING? Sub sampling has a great variety of applications, which go far beyond the immediate scope of sample surveys. Whenever any process involves chemical,physical,or biological tests that can be performed on a small amount of material, it is likely to be drawn as a subsample from a larger amount that is itself a sample. The principal advantage of 2-stage sampling is that it is more flexible than 1-stage sampling. It appears to be more efficient. There’s a balance between precision & cost 4

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MEAN & VARIANCE IN 2-STAGE SAMPLING In finding the mean & variance of this example, averages must be taken over all samples that can be generated by this 2-stage process. The way of calculating this average is first to average the estimate over all second-stage selections that are drawn from the set of “n” units that are selected, then average overall possible selections of “n” units by the plan. 5

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The following steps are used for calculating the mean & variance of the sampling- Yij=value obtained for the jth subunit in the ith primary unit 1.To calculate ȳi=∑yij/m (sample mean per subunit in the ith primary unit) 2. To calculate y⁼=∑ȳ̄i/n (over-all sample mean per subunit) After calculating these, calculation for various is done using following formulae- S1²=∑(Ȳi-Y⁼)²/N-1 (variance among primary units means) S2²=∑ ∑(yij-Ȳi)²/N(M-1) (variance among subunits within primary units) j=1 to m i=1 to n i=1 to N i=1 to N j=1 to m 6

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THREE STAGE SAMPLING The process of sub sampling can be carried to a third stage by sampling the subunits instead of enumerating them completely. E.g.- the village is a convenient sampling unit(primary unit),a field in the village is a subunit(secondary unit),and the part of that field sampled for the determination of yield per acre is known to be the tertiary unit, i.e. the subunit is itself sampled. 7

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EXAMPLE: If we take population as a sample. The population contains N first stage units, each with M second stage units ,each of which has K-third stage units Let yiju be the value obtained for the uth third stage unit in the jth second stage unit drawn from the ith primary unit. The relevant population means per third stage unit are as follows: Ȳij=∑yiju/K, Y⁼i=∑ ∑ yiju/MK, Ȳ⁼=∑∑∑yiju/NMK u to K j to M u to K i to N u to K 8

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The following population variances are required. S1²=∑(Y⁼i-Y⁼̄)²/N-1 S2²=∑∑(Ȳij-Y⁼)²/N(M-1) S3²=∑∑∑(yijk-Ȳij)²/NM(K-1) i to N i to N j to M i to N u to K 9

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ADVANTAGES OF MULTISTAGE SAMPLING 1.Is done when the sampling unit is quite large. 2.A frame is constructed for every sample unit,i.e there must be a frame that describes all the primary units in the universe. 3.At every successive stage the frames become more & more detailed as the size of the sampling unit becomes smaller & smaller. 4.Frames are prepared for next stage only for the units that have already fallen into the sample. 10

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Thanks for your cooperation 11