Stratified sample

A sample which is broken up into several samples of smaller sizes by certain distinguishing marks (characteristics). Let each element of some sample of size $N \geq 2$ possess one and only one of $k \geq 2$ possible marks. Then the original sample can be broken into $k$ samples of sizes $n _ {1} \dots n _ {k}$, respectively $( n _ {1} + \dots + n _ {k} = N)$:

$$\begin{array}{c} X _ {11} \dots X _ {1n _ {1} } , \\ X _ {21} \dots X _ {2n _ {2} } , \\ {} \dots \dots \dots \\ X _ {k1} \dots X _ {kn _ {k} } , \\ \end{array}$$

where the $i$-th sample $X _ {i1} \dots X _ {in _ {i} }$ contains only those elements of the original sample which have the mark $i$. As a result of this decomposition, the original sample becomes stratified into $k$ strata $X _ {i1} \dots X _ {in _ {i} }$, $i = 1 \dots k$, where the $i$-th stratum contains information about the $i$-th mark. This notion gives rise, for example, to realizations of the $X$-component of a two-dimensional random variable $( X, Y)$ whose second component $Y$ has a discrete distribution.

References

[1]	S.S. Wilks, "Mathematical statistics" , Wiley (1962)

References

[a1]	W.G. Cochran, "Sampling techniques" , Wiley (1977)