Editing Variance (section)

=====Sum of uncorrelated variables with random sample size=====
There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size {{math|N}} is a random variable whose variation adds to the variation of {{math|X}}, such that,<ref>Cornell, J R, and Benjamin, C A, ''Probability, Statistics, and Decisions for Civil Engineers,'' McGraw-Hill, NY, 1970, pp.178-9.</ref>
<math display="block">\operatorname{Var}\left(\sum_{i=1}^{N}X_i\right)=\operatorname{E}\left[N\right]\operatorname{Var}(X)+\operatorname{Var}(N)(\operatorname{E}\left[X\right])^2</math>
which follows from the [[law of total variance]].

If {{math|N}} has a [[Poisson distribution]], then <math>\operatorname{E}[N]=\operatorname{Var}(N)</math> with estimator {{math|n}} = {{math|N}}. So, the estimator of <math>\operatorname{Var}\left(\sum_{i=1}^{n}X_i\right)</math> becomes <math>n{S_x}^2+n\bar{X}^2</math>, giving <math>\operatorname{SE}(\bar{X})=\sqrt{\frac{{S_x}^2+\bar{X}^2}{n}}</math>
(see [[Standard error#Standard_error_of_the_sample_mean|standard error of the sample mean]]).