I have recorded N, Mean, SD of a variable (X) from N studies. The size of ni varies from 4 to 784. The mean ranges from 22 to 175 and SD ranges from 0.7 to 134.9. With stratification over ni. I see that X follows normal distribution. But the distribution of SD and CV are not following Normal distribution. Both, SD and CV are positively skewed.
Now I want to have a good estimate the mean of that variable under above circumstances. Also I wish to estimate the SD of new estimate.
A variable can be characterised by Mean and SD when it is normally distributed.
In my case, the mean of means from several studies follows Normal distribution but the SD from those studies do not follow normal distribution.
How to handle this situation when my interest is to have combined estimate of mean and SD of variable? If weighted mean is required then How?
It is theoretically expected that the mean of means be normally distributed. Your observation is therefore correct. The SD's are not expected to follow a normal distribution.
The combined estimate of the mean is the weighted mean of the means. The combined estimate of the SD is the square root of the pooled variance.
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