Based on this very good post How can I calculate margin of error in a NPS (Net Promoter Score) result? I've pulled together the detailed calculations to perform the test for the generalised NPS problem.
I suspect that this question is lower order than generally posted here but my last true stats course was 20 years ago and I'm hoping that someone will validate the steps.
To determine if there is a significant (95%) probability of a real difference between two sets of NPS scores:
#P is the number of Promoters #N is the number of Neutrals #D is the number of Detractors. #T = #P + #N + #D
First calculate NPS in the normal way:
NPS = #P/#T - #D/#T
Now determine the Variance of the sample NPS:
Var(NPS) = (1 – NPS) ^ 2) * #P/#T + (0-NPS) ^ 2 * #N/#T + (-1-NPS) ^ 2 * #D/#T)
Now calculate the Margin of Error (MoE) for the sample:
MoE = SQRT(Var(NPS)) / SQRT(#T)
If you do this for both of your samples you will end up with two MoE, one for each sample.
To determine if your score probably (95%) changed:
If ABS(NPS1 – NPS2) > 2 * SQRT (MoE1 ^ 2 + MoE2 ^ 2) then the difference is real