weighted standard error of the mean


I am trying to calculate some statistical properties pertaining to mobility of cargo inside cells for my research, but due to my poor statistical skills I am in need of a little guidance. I think the short version of my question is how to calculate the standard error of the mean with samples that are weighted differently.

In more detail...

I am studying the speed distribution of particles moving inside of cells. It turns out that a useful way to study the speed distribution for my experiments is to plot the survival function (1-CD or cumulative distribution).

Now I want to estimate the error in the survival functions. In order to do this, I simply repeated the experiment several times (5-10). For each experiment I calculated a survival function. From there I calculate a weighted mean SF, where the weights are proportional to the number of samples in each SF. (One experiment might have 100 particles, while another might have 500, so I figured I should give them different weights.)

I found on Wikipedia an equation for calculating the weighted variance (URL below). From the weighted variance I now want to calculate the standard error of the mean so that I can compare one batch of control experiments to another batch of experiments where a 'motion-killing drug' has been applied to the cells. This way I can determine whether differences between the two batches of experiments are meaningful.

For equally weighted samples I know the equation is simply SEM = SD/(N^0.5). Any ideas how (or if) this equation should be modified when the samples are weighted?


The definition of weighted standard error


The question you are asking is important but difficult to be answered since there is no consensus on the definition of weighted standard error (WSE). We have the definitions on both weighted mean and weighted standard deviation (WSD) but there is no definition for the WSE; thus no weighted confidence interval (WCI) for a weighted mean.

Some people use the WSD and sample size n to define the WSE; but others use WSD and a sum of weights to define the WSE.

In my opinion, the latter one is correct because the sample size n has two following basic properties in the regular statistics:

1) it is a sum of weights;

2) it is a particular case under the condition of all weights being equal to 1.

Therefore, all regular statistics can be generalized with the sum of weights.

I did a mathematical proof on the two properties in my paper, which can be found in the 2007 JSM Proceedings. You can use my name "Ligong Chen" to search in the proceedings.

If you need help, please email me at chenlgyq@*******.com
or visit my personal website at: http://cotglti.spaces.live.com

Thank you!
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