# Estimating a population total from a stratified simple random sample

I have a stratified population. I want to estimate a population total $$T$$ from a stratified simple random sample. I have two strategies:

1. I compute $$\displaystyle T=\sum_h N_h\bar x_h$$ where $$x_h$$ is the sample mean of the stratum $$h$$ and $$N_h$$ is the number of elements of the population in stratum $$h$$.

2. I compute $$T$$ as the sum of all observations where the sampled elements are given their actual value and each remaining non sampled element is given the value of the sample mean of the stratum to which it belongs.

I want to know which strategy is better to estimate the population total $$T$$. Thanks!

• Could you provide an example in which the two procedures produce different values?
– whuber
Commented May 30, 2022 at 19:48
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented May 30, 2022 at 21:34

#### Those two sums are mathematically equivalent

To analyse your proposal, I will call your two totals $$T_1$$ and $$T_2$$ respectively. Suppose we index the population values as $$x_{h,i}$$ where $$h=1,...,H$$ is the stratum and $$i = 1,...,N_h$$ is the corresponding value number. We will denote the imputed value for the second total as:

$$\tilde{x}_{h,i} = \begin{cases} x_{h,i} & & \text{for } 1 \leqslant i \leqslant n_h, \\[6pt] \bar{x}_h & & \text{for } n_h < i \leqslant N_h. \\[6pt] \end{cases}$$

\begin{align} T_1 &\equiv \sum_{h=1}^H N_h \bar{x}_h, \\[6pt] T_2 &\equiv \sum_{h=1}^H \sum_{i=1}^{N_h} \tilde{x}_{h,i}. \\[6pt] \end{align}
\begin{align} T_2 &= \sum_{h=1}^H \sum_{i=1}^{N_h} \tilde{x}_{h,i} \\[6pt] &= \sum_{h=1}^H \Bigg[ \sum_{i=1}^{n_h} x_{h,i} + \sum_{i=n_h+1}^{N_h} \bar{x}_h \Bigg] \\[6pt] &= \sum_{h=1}^H \Bigg[ n_h \bar{x}_h + (N_h-n_h) \bar{x}_h \Bigg] \\[6pt] &= \sum_{h=1}^H N_h \bar{x}_h \\[6pt] &= T_1. \\[6pt] \end{align}
• Thanks it is very clear. What if I want to compute substotals of the variable, for example I want the subtotal for each population stratum or any other breakdown of the variable, which one do I use when I compute the subtotal: I give the sampled units their actual values $x_{h,i}$ or the sample mean of the stratum $\bar x_h$? Commented May 31, 2022 at 5:46