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  "body": "\n## Definition\n\nLet $U$ be a finite population of $N$ units, and let $U_1, \\dots, U_H$ be a [[partition|08QMKTWF9T]] of $U$ into nonempty strata of sizes $N_1, \\dots, N_H$.\nChoose sample sizes $n_1, \\dots, n_H$ with $1 \\le n_h \\le N_h$.\n*Stratified (simple random) sampling* with these sizes draws, for each $h$, a [[simple random sample|Y0PTCJBPEA]] $S_h$ of size $n_h$ from $U_h$.\nThe $H$ draws are [[independent|0GQ938GDE0]].\nThe sample is $S = S_1 \\cup \\cdots \\cup S_H$.\n\n## Remarks\n\nFor values $y_i$ attached to the units, the stratified estimator of the population mean is $\\bar y_{st} = \\sum_{h=1}^{H} \\frac{N_h}{N} \\bar y_h$, where $\\bar y_h$ is the sample mean in $S_h$.\nIt is unbiased, with variance $\\sum_h \\big(\\frac{N_h}{N}\\big)^2 \\big(1 - \\frac{n_h}{N_h}\\big) \\frac{S_h^2}{n_h}$, where $S_h^2$ is the population variance of the $y_i$ in $U_h$ with divisor $N_h - 1$.\n*Proportional allocation* takes $n_h$ proportional to $N_h$.\n*Neyman allocation* takes $n_h$ proportional to $N_h S_h$, which minimizes the variance for a fixed total $n = \\sum_h n_h$.\n\nExample: $U = \\set{1, 2, 3, 4}$ with strata $U_1 = \\set{1, 2}$ and $U_2 = \\set{3, 4}$, and $n_1 = n_2 = 1$.\nThe possible samples are $\\set{1, 3}$, $\\set{1, 4}$, $\\set{2, 3}$ and $\\set{2, 4}$, each with probability $1/4$.\n\nNon-example: a simple random sample of size $2$ from the whole of $U$.\nIt can be $\\set{1, 2}$, with no unit from $U_2$, so it is not a stratified sample with $n_1 = n_2 = 1$.\n",
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