Solved – Standard Error for Sum

Question

I am designing an algorithm for a stratified sampling on a population and then I want to find out what is the error bound for 95% confidence interval, for different sample statistic such as sum of the samples. For this purpose, I need to find Standard Error for sum. I cannot find any information on Standard error other than for mean and proportion. My purpose is SE for sum.. Any information on the formula for SE for SUM, or any links which gives an idea is highly appreciated ! (I have heard SE calculation differs for different sampling techniques, any info about that would be great too).

Answer 1

If you assume that $X_1, ..., X_n$ is a sample from a Gaussian distribution $N(\mu, \sigma^2)$, then this theorem stipulates that a linear combination of the data, $\sum_i c_i X_i$, has distribution $$Y := \sum_{i=1}^{n} c_i X_i \sim N\left( \sum_{i} c_i \mu, \ \sum_{i} c_{i}^2\sigma^2 \right)$$ In your example, $c_i =1 \ \forall i$, so defining $Y$ as the sum, $$Y \sim N\left(n\mu, n\sigma^2 \right)$$ which means that the standard error (i.e. the standard deviation of the sampling distribution of $Y$) is $\sqrt{n}\sigma$.

Now, this example is Gaussian-specific, but in general, the standard error of a statistic is the standard deviation of its sampling distribution. See this discussion for a good breakdown of how standard deviation and standard error differ, and this answer for general standard error discussion.