Calculating the Net Promoter Score for poll data on a 1-5 scale From Wikipedia

The Net Promoter Score is obtained by asking customers a single
  question on a 0 to 10 rating scale, where 10 is "extremely likely" and
  0 is "not at all likely": "How likely is it that you would recommend
  our company to a friend or colleague?" Based on their responses,
  customers are categorized into one of three groups: Promoters (9–10
  rating), Passives (7–8 rating), and Detractors (0–6 rating). The
  percentage of Detractors is then subtracted from the percentage of
  Promoters to obtain a Net Promoter score (NPS). NPS can be as low as
  -100 (everybody is a detractor) or as high as +100 (everybody is a promoter).

Now we want to scale it down to 1-5 for our purposes, because frankly our users will get confused on the 0-10 scale. Now we are considering 1,2,3 to be detractors , 4 to be passive and 5 to be a promoter. Now In the Standard 0-10 scale, it is actually a 11 point scale – so promoters (9,10) is 2/11th of the distribution which is 18%. In the 1-5 scale we use promoters (5) is 1/5 which is 20%. So more chances of getting a promoter score. By same logic, 5 point scale will also show less detractors than 0-10 point scale (3/5=60% vs 7/11 = 63.6%). So a positive skew looks possible mathematically. But the fact is a user has only 1 option to choose from on the 1-5 scale to be a promoter, but 2 on the 1-10 scale. So someone who might be a 9 on a 10 scale might be a 4 on the 5 scale and become a passive and never show up and kill our NPS score to go to -100(if no one votes 5 , and a lot of people vote 4) . Three questions


1.) Which one of the two math arguments should be considered for scaling down
2.) What should be the right passive pivot for the 1-5 scale?

3.) Are there other tests that can be done on this data to get an accurate measure of customer feedback?

 A: NPS isn't statistics but an attempt to measure perception of a company, product or service. So too, my answer isn't statistical but psychological or practical. 
Don't scale down. 
NPS' scale is deliberately 0 - 10 to allow both 9 and 10 to be promoters. There are many people who will refuse to give a "perfect" score but who are avid promoters. So you need both 9 and 10. 
If you must go 1 to 5, then you need 4 and 5 to be promoters. That means 40% are now in the promoter range rather than 18%, but NPS isn't concerned with that*. 
At the same time, 0 - 6 detractors become 1 - 3 by scale and we're left with no "passives". So do we go with 1 - 2 then? No. Don't do that. Just get rid of passives. You need more detractor options than promoter options because detractors carry more weight. To get a net-positive score you need significantly more promoters. 
Imagine you go to a conference and you speak to five people about some company. Two say "it's great". Two say "it sucks". With the fifth person scoring "3" (passive) you now have equal promoters and detractors. But psychologically you're much more likely to feel negative about the company. Is a neutral score (40% - 40% = 0 NPS) therefore valid? No. 
So let's put person number five's score of '3' in the detractors. Now our score is 40% - 60% = -20 NPS. A negative score. Much more representative of actual results of the opinions of those surveyed.  
* However you do it, the concept remains. If you change it from 0-10 to 1-5, you can't compare to a competitor that uses 0-10 and just scale. There's more psychology than statistics at play here. What you can (and should) do is compare it to your results using 1-5 from your last survey. And that's what ultimately matters: has our market perception gone up or down?
