How to convert sampling sampling variances for a meta-analysis using raw means as effect sizes?

I am conducting a meta-analysis using raw mean scores as effect sizes (i.e., reported test scores on a certain standardized test).

I found a calculation formula for sampling variances provided by, e.g., Card (2012, p. 150), Hox (2010, p. 209), and also [Calculation for the estimated sampling variance of individual group raw mean.

SEx = Sx/√N

• where s is the standard deviation of variable X, and N is the sample size.

Especially, Card's book, Applied Meta-analysis for social science research was very helpful providing an equation for transforming scores between two different scales.

Card (2012, p. 148)

X2 = ((X1 - Min1)((Max2 - Min2)/(Max1 - Min1))+Min2

• where, X2 is the equivalent score on the second scale
• X1 is the score on the first scale that you wish to transform.
• Min1 is the lowest possible score on the first scale.
• Max1 is the highest possible score on the first scale.
• Min2 is the lowest possible score on the second scale.
• Max2 is the highest possible score on the second scale.

However, the book does not provide how to calculate sampling variances when ESs were transformed into a standardized value.

I would really appreciate it if you could let me know how sampling variances could be calculated for this standardized effect size. If possible, references that I can cite would be much appreciated.

• SE(X2) = SE(X1)(Max2 - Min2)/(Max1 - Min1) – user158565 Aug 10 at 2:04
• Thank you for your response, @user158565! Could you let me know how you find the formula? I am wondering if you know any reference that I can cite. Thank you in advance! – user8460166 Aug 10 at 4:19