There is a rich literature on calculating variance associated with each influence. This is a major concern when conducting a meta-analysis and the methods include evaluating the degree of heterogeneity across the studies and assessing whether the mean is a reasonable typical measure. To mitigate this issue, it is important that the focus is then the search for moderators or mediators to better help explain what is happening across the studies. It is this focus that was adopted by John Hattie in the Visible Learning (2009) and subsequent publications and the following articles and book provide an excellent technical understanding of the approaches that can be used, and those which are presented as being optimal.
Takkouche, B., Khudyakov, P., Costa-Bouzas, J., & Spiegelman, D. (2013). Confidence intervals for heterogeneity measures in meta-analysis. American Journal of Epidemiology, kwt060.
Borenstein, M., Hedges, L. V., Higgins, J. P., & Rothstein, H. R. (2011). Introduction to meta-analysis. John Wiley & Sons.
Huedo-Medina, T. B., Sánchez-Meca, J., Marin-Martinez, F., & Botella, J. (2006). Assessing heterogeneity in meta-analysis. Psychological methods, 11(2), 193.