Customer experience metrics
NPS margin of error and significance testing
The margin of error of a Net Promoter Score is z × SE, where SE = 100 × √{[pP(1 − pP) + pD(1 − pD) + 2·pP·pD] / n} and pP, pD are the promoter and detractor shares. It peaks when half the sample are promoters and half detractors: ±9.8 points at 95% confidence with 400 respondents. In the worked example below, 600 respondents give ±6.3 points. To test whether two independent NPS results differ, divide their difference by √(SE₁² + SE₂²) and compare with 1.96.
How results are checked
Named statistical review pending. Selected automated reference cases are published on the validation page.Why NPS needs its own formula
NPS is not a single proportion. Each respondent is a promoter (9–10), a passive (7–8) or a detractor (0–6), and the score is the promoter share minus the detractor share. Both shares come from the same respondents and are negatively related (nobody can be both), so the variance of their difference is pP(1 − pP) + pD(1 − pD) + 2·pP·pD, divided by n. Treating NPS as one percentage, or adding the two shares' margins, gets the uncertainty wrong.
Passives matter even though they do not appear in the score: the more respondents are passive, the smaller the variance and the narrower the interval for the same n.
Comparing two NPS results
For two independent groups (two segments, or two waves with different respondents), the standard error of the difference is √(SE₁² + SE₂²) and z = (NPS₂ − NPS₁) / SEdiff. Testing many segments against each other multiplies the chance of a false alarm, so either decide the comparisons in advance or correct for multiple comparisons. A panel that answers twice needs a paired analysis instead.
The method
Formula
NPS = 100 × (pP − pD) · SE = 100 × √{[pP(1 − pP) + pD(1 − pD) + 2·pP·pD] / n} · MOE = z × SE
z = (NPS₂ − NPS₁) / √(SE₁² + SE₂²) · nrequired = z² × [pP(1 − pP) + pD(1 − pD) + 2·pP·pD] / MOE²
where
- p_P, p_D
- shares of promoters (9–10) and detractors (0–6) among valid answers
- n
- valid responses; for weighted data the Kish effective sample size (Σw)² / Σw²
- z
- 1.645, 1.960 or 2.576 for 90%, 95% or 99% confidence
- MOE
- margin of error as a proportion (0.05 for ±5 points) in the sample-size formula
Worked example
Sample data: the 0–10 recommend question in the synthetic snack-brand tracker that the NPS calculator loads with “Use sample data” (600 generated respondents, not real customers).
Margin of error. 25.0% are promoters and 39.5% detractors, so NPS = −14.5. The variance term is 0.6240, the standard error is √(0.6240 / 600) × 100 = 3.22 points, and the 95% interval is −20.8 to −8.2: a margin of error of ±6.3 points.
Significance test. Women score −8.2 (n = 306, SE 4.66) and men −21.0 (n = 276, SE 4.61). The gap is 12.8 points with SE √(4.66² + 4.61²) = 6.55, so z = 1.961 and two-sided p = 0.0499. That is just under 0.05: the 95% interval for the gap, 0.0 to 25.7 points, barely excludes zero. A 13-point gap between groups of about 300 is at the edge of what this sample can detect; report it with the interval, not as a firm finding.
Sample size. With the same mix of promoters and detractors, a ±5-point margin at 95% needs n = 1.96² × 0.6240 / 0.05² ≈ 959 respondents, and ±3 points needs about 2,664.
Using it well
When to use it
Use it when
- Use: Reporting an NPS with its uncertainty on a dashboard or in a tracker.
- Use: Deciding whether a change between waves, or a gap between segments, is more than sampling noise.
- Use: Planning how many completes a study needs for a given NPS precision.
Not the right tool when
- Avoid: The same respondents answered both times: the independent-samples test overstates the uncertainty of the change.
- Avoid: The score is not the 0–10 recommend question; NPS cut-offs do not apply to other scales.
- Avoid: The sample is not a probability sample of the customers you report on: the interval covers sampling error only, not coverage or non-response bias.
How to interpret it
Rule of thumb
An NPS movement smaller than the margin of error is not evidence of change. Report the interval with every score, test planned comparisons, and treat a result with p close to 0.05 as a lead to confirm in the next wave.
Next step: calculate your NPS interval
The free NPS calculator gives NPS with its standard error and 90, 95 or 99% Wald and adjusted-Wald intervals from counts or a respondent file, compares two groups with the z-test above, and breaks NPS down by segment with each segment tested against the rest of the sample. Weighted files use the Kish effective sample size. It runs in your browser with no account.
Questions
Frequently asked questions
Why is the NPS margin of error bigger than for a percentage?
NPS is the difference of two shares from the same respondents, promoters minus detractors, so the uncertainty of both shares adds up. On a 0–10 base where promoters and detractors are each common, the standard error of NPS in points can be up to twice that of a single percentage from the same sample.
What sample size do I need for a ±5 point NPS margin?
It depends on the mix of promoters, passives and detractors. Use n = 1.96² × [pP(1 − pP) + pD(1 − pD) + 2·pP·pD] / 0.05². The worst case, half promoters and half detractors, needs about 1,537 respondents; with fewer of both, as in the worked example, it is closer to 1,000.
How do I test whether NPS changed between two waves?
For two independent samples, divide the difference in NPS by the square root of the sum of the two squared standard errors and compare the z value with 1.96 for 95% confidence. If the same respondents answered both waves (a panel), this test is too conservative; a paired analysis of each respondent's change is needed.
Should I use the adjusted Wald interval for NPS?
For small samples or extreme scores, yes. Adding three pseudo-respondents (three quarters of a promoter, one and a half passives, three quarters of a detractor) before computing the interval gives better coverage than the plain Wald interval (Rocks, 2016). With a few hundred respondents the two are nearly identical.
How does weighting change the NPS margin of error?
Weights make the effective sample smaller than the number of interviews. Replace n with the Kish effective sample size, (Σw)² / Σw², in the standard error. Unequal weights always widen the interval; they never narrow it.
Sources and next steps
References & further reading
- [1]Reichheld, F. F. (2003). The one number you need to grow. Harvard Business Review, 81(12), 46–54.
- [2]Rocks, B. (2016). Interval estimation for the “net promoter score”. The American Statistician, 70(4), 365–372.
Try it on your own data — free, no signup
Upload a CSV, Excel, SPSS, Stata, SAS, R, Parquet or JSON file. Everything runs in your browser; your file never leaves your device.
Open the workspace →