I am developing a new psychometric scale consisting of a several dichotomous items taken from a larger item bank. The p values (item means) are known. Now I need to calculate the expected mean and standard deviation of the scale after having picked the items. is there a formula, or well known procedure for it? I am looking for the best predicted mean and SD. I know there will allways be uncertainty about the true mean and SD, but for now that's not my main concern.
Many thanks in advance.
Edit. The scale is a lineair combination (just the sum) of the dichotomous items.
edit. This is an example of all the information I have of the picked items (=variables). Unfortenately I don't have the covariances.
M_item SD_item 0.8622312 0.344657159720439 0.3368913 0.472647386520131 0.5976337 0.490375020391853 0.6673509 0.471162048842211 0.8394329 0.367131184180246 0.7136281 0.452065299365468 0.7948207 0.403832582701681 0.7589461 0.427722944550313 0.8300466 0.375591855380864