I don't know if this is possible, but I would like to know if it is and if so how to arrive at the solution:

If I have a list of numbers of which size is known, but members are not known, their average and variance or standard deviation is known, from this an arbitrary number x is removed from the list. I can calculate the new average of the restering list fine but I'm puzzled at how to arrive at the new variance, or if it is possible at all in a general way to do it.

Another way, perhaps better way of formulating the problem is that I have a list of numbers and add x (which is known, because it is chosen) to form a new list, of which variance, average, and size is known, and from this somehow figuring out what the variance is of the original list is.

Intuitively it would seem that the farther away from the average I pick the number x, the smaller the variance will be, and that suggests that it should be possible to estimate it somehow.

If I cannot do this, then is it possible to on the basis of assuming that the numbers roughly follow a normal distribution, somehow approximate the variance?