If I understand correctly, you are trying to compute a partial average of field quantities of three-dimensional data. This can be expressed as a partial integration:
\bar{v}(\xi, \eta) \equiv \frac{1}{\zeta_f-\zeta_0}\int^{\zeta_f}_{\zeta_0} v(x,y,z)d\zeta
For your specific case, I believe you want to average this directly along the z-coordinate axis, which makes things a bit easier:
\bar{v}(x, y) = \frac{1}{z_f-z_0}\int^{z_f}_{z_0}v(x,y,z)dz
For the accumulation (again, assuming I’ve understood correctly what you’re trying to do), you can simply leave off the fraction.
The easiest way to do this in Paraview is probably with a programmable filter. I did something very similar here: Select columns of structured data - ParaView Support - ParaView, where I computed the accumulation of density based on a Wavelet Source.