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).

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    $\begingroup$ The mean and the sum are usually related by a constant multiple--typically, the mean is $1/n$ times the sum. Since it is obvious that their standard errors will be related by the same multiple, people rarely stop to give explicit formulas for both sums and means: one formula is good enough. $\endgroup$
    – whuber
    Aug 3, 2015 at 17:24
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    $\begingroup$ Describe your sampling i more detail, show equations $\endgroup$
    – Aksakal
    Aug 3, 2015 at 17:25

1 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.

  • $\begingroup$ If Standard Err, SE for mean = Standard Deviation/sqrt(Sample size) [information from a scientific paper], then is the SE for Sum = sqrt(Sample Size) * Standard Deviation ? I have also seen the latter formula being used for a scientific tool.. but these contrasts with the above result.. The population being just some random independant numbers .. Please correct me if wrong $\endgroup$
    – drk
    Aug 4, 2015 at 6:31
  • $\begingroup$ @user3218207 yes, the standard error of a sum of iid random variables is $\sigma\sqrt{n}$ if each RV, $X_i$, has variance $\text{Var}(X_i)=\sigma^2$, as pointed out in the last link above. $\endgroup$
    – call-in-co
    Aug 4, 2015 at 15:15
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    $\begingroup$ The formula above applies to a simple random sample froma population with known $\mu$ and $\sigma$. The question is about stratified sampling of an arbitrary population, whether with or without replacement is not specified. Both mean and SD are unknown and must be estimated. The theory and formulas are given in every sampling text. $\endgroup$ Aug 11, 2015 at 21:32

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