I ran into the same issue recently. The following is what I found:

Unlike a simple random sample with equal weights, there is no widely accepted definition of standard error of the weighted mean. These days, it would be straight-forward to do a bootstrap and obtain the empirical distribution of the mean, and based on that estimate the standard error.

What if one wanted to use a formula to do this estimation?

The main reference is this paper, by Donald F. Gatz and Luther Smith, where 3 formula based estimators are compared with bootstrap results. The best approximation to the bootstrap result comes from Cochran (1977):

$(SEM_w)^2={\dfrac{n}{(n-1)(\sum {P_i})^2}}[\sum (P_i X_i-\bar{P}\bar{X}_w)^2-2 \bar{X}_w \sum (P_i-\bar{P})(P_i X_i-\bar{P}\bar{X}_w)+\bar{X}^2_w \sum (P_i-\bar{P})^2]$

The following is the corresponding R code that came from this R listserve thread.

weighted.var.se <- function(x, w, na.rm=FALSE) # Computes the variance of a weighted mean following Cochran 1977 definition { if (na.rm) { w <- w[i <- !is.na(x)]; x <- x[i] } n = length(w) xWbar = weighted.mean(x,w,na.rm=na.rm) wbar = mean(w) out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w-wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2)) return(out) } 

Hope this helps!

I ran into the same issue recently. The following is what I found:

Unlike a simple random sample with equal weights, there is no widely accepted definition of standard error of the weighted mean. These days, it would be straight-forward to do a bootstrap and obtain the empirical distribution of the mean, and based on that estimate the standard error.

What if one wanted to use a formula to do this estimation?

The main reference is this paper, by Donald F. Gatz and Luther Smith, where 3 formula based estimators are compared with bootstrap results. The best approximation to the bootstrap result comes from Cochran (1977):

$(SEM_w)^2={\dfrac{n}{(n-1)(\sum {P_i})^2}}[\sum (P_i X_i-\bar{P}\bar{X}_w)^2-2 \bar{X}_w \sum (P_i-\bar{P})(P_i X_i-\bar{P}\bar{X}_w)+\bar{X}^2_w \sum (P_i-\bar{P})^2]$

The following is the corresponding R code that came from this R listserve thread.

weighted.var.se <- function(x, w, na.rm=FALSE) # Computes the variance of a weighted mean following Cochran 1977 definition { if (na.rm) { w <- w[i <- !is.na(x)]; x <- x[i] } n = length(w) xWbar = weighted.mean(x,w,na.rm=na.rm) wbar = mean(w) out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w-wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2)) return(out) } 

Hope this helps!

I ran into the same issue recently. The following is what I found:

Unlike a simple random sample with equal weights, there is no widely accepted definition of standard error of the weighted mean. These days, it would be straight-forward to do a bootstrap and obtain the empirical distribution of the mean, and based on that estimate the standard error.

What if one wanted to use a formula to do this estimation?

The main reference is this paper, by Donald F. Gatz and Luther Smith, where 3 formula based estimators are compared with bootstrap results. The best approximation to the bootstrap result comes from Cochran (1977):

$(SEM_w)^2={\dfrac{n}{(n-1)(\sum {P_i})^2}}[\sum (P_i X_i-\bar{P}\bar{X}_w)^2-2 \bar{X}_w) \sum (P_i-\bar{P})(P_i X_i-\bar{P}\bar{X}_w)+\bar{X}^2_w \sum (P_i-\bar{P})^2]$

The following is the corresponding R code that came from this R listserve thread.

weighted.var.se <- function(x, w, na.rm=FALSE) # Computes the variance of a weighted mean following Cochran 1977 definition { if (na.rm) { w <- w[i <- !is.na(x)]; x <- x[i] } n = length(w) xWbar = weighted.mean(x,w,na.rm=na.rm) wbar = mean(w) out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w-wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2)) return(out) } 

Hope this helps!

I ran into the same issue recently. The following is what I found:

Unlike a simple random sample with equal weights, there is no widely accepted definition of standard error of the weighted mean. These days, it would be straight-forward to do a bootstrap and obtain the empirical distribution of the mean, and based on that estimate the standard error.

What if one wanted to use a formula to do this estimation?

The main reference is this paper, by Donald F. Gatz and Luther Smith, where 3 formula based estimators are compared with bootstrap results. The best approximation to the bootstrap result comes from Cochran (1977):

$(SEM_w)^2={\dfrac{n}{(n-1)(\sum {P_i})^2}}[\sum (P_i X_i-\bar{P}\bar{X}_w)^2-2 \bar{X}_w \sum (P_i-\bar{P})(P_i X_i-\bar{P}\bar{X}_w)+\bar{X}^2_w \sum (P_i-\bar{P})^2]$

The following is the corresponding R code that came from this R listserve thread.

weighted.var.se <- function(x, w, na.rm=FALSE) # Computes the variance of a weighted mean following Cochran 1977 definition { if (na.rm) { w <- w[i <- !is.na(x)]; x <- x[i] } n = length(w) xWbar = weighted.mean(x,w,na.rm=na.rm) wbar = mean(w) out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w-wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2)) return(out) } 

Hope this helps!