The example wall section at right consists of two different cross sections: (A) where there are no 2x4 studs: it's sheetrock-insulation-plywood-siding, and (B) the section where there are studs: it's sheetrock-2x4-insulation-2x4-plywood-siding. In this example, the 2x4s are 24" apart, that means every 24" section of wall consists of 22.5" of assembly A and 1.5" of assembly B.
The R value for section A is: .6 (sheetrock@ R1 per inch) + 33.3 (cellulose@ R3.7 per inch) + .5 (plywood@ R1 per inch) + .5 (siding+air barrier: estimate )=R34.9.
The R value for section B is: .6 (sheetrock@ R1 per inch ) +3.5(2x4)+7.4(cellulose) + 3.5(2x4) + .5 (plywood) + .5 (siding) = R16.
To get the R value of the complete wall, we add up the U values of each section multiplied by what percentage of the overall assembly they represent, and then take the inverse. For our sample wall, section A is 94% (ie 22.5" out of 24"), and so section B is 6%. The basic formula is:
Uwall = Ua*Pa + Ub+Pb + Uc*Pc +
.... where Ux is the U value of a section and Px is the section's percentage of the whole.
For our wall, Uwall = (1/34.9)*.94+(1/16)*.06 = .0307, which is a R value of about 32.5.Standard framing factors are much higher than the sample wall section shown above--the range from 12% to 20%--that is two to three times more than the sample above. While this does reduce the whole wall R-value, the effect isn't terrible:
Doubling the framing to 12% results in: (1/34.9)*.88+(1/16)*.12=R30.6
Increasing the framing to 20% results in : (1/34.9)*.8+(1/16)*.2=R28.2