CSAT and NPS calculator with margin of error

How to use it

Choose CSAT or NPS. For CSAT, enter the satisfied responses under your rule (for example, 4 and 5 on a 1-to-5 scale) and the total of valid responses; if you know the eligible contacts, the tool also works out the response rate. For NPS, enter how many people are promoters, passives and detractors. Block B is optional and lets you compare two periods or two segments, such as bot-only chats and chats with an agent. The target margin shows how many responses you would need to measure with that precision.

What it calculates: formulas and assumptions

CSAT = satisfied / valid responses, with neutral and dissatisfied responses in the denominator, as in the guide to the CSAT survey after a phone call. The response rate is responses / eligible contacts.

The 95% interval uses the Wilson method, which the NIST/SEMATECH e-Handbook sets out as an alternative to the usual normal approximation (Wald, p ± 1.96 × √(p(1 − p)/n)) and which, according to that page, Agresti and Coull and Brown, Cai and DasGupta recommend. Wald is symmetrical and can give a negative lower limit, which NIST regards as the mark of an inferior method; with 10 out of 10 satisfied its margin would be zero, as if there were no uncertainty. NIST warns that with small samples the symmetrical limits of the normal approximation may not be accurate enough. Wilson centres the interval on (p + z²/2n) / (1 + z²/n), never leaves 0-100% and is asymmetrical when the proportion is close to an end of the scale. The tool also shows the Wald margin so you can compare it with published figures.

NPS = % promoters − % detractors, following the definition from Bain & Company (promoters 9-10, passives 7-8, detractors 0-6). It is expressed in points, from −100 to 100.

Its margin is not obtained by adding two proportion margins, because promoters and detractors come from the same multinomial sample and are correlated. If each response scores +1 (promoter), 0 (passive) or −1 (detractor), NPS is their mean. Its variance per response is E[X²] − E[X]² = p + d − (p − d)², where p and d are the shares of promoters and detractors. This equals p(1 − p) + d(1 − d) + 2pd. The margin is 1.96 × √(var / n) × 100 points, and the sample size for a margin E is n = 1.96² × var / E². The maximum variance is 1 (half promoters, half detractors); for CSAT, the worst case is p = 50%.

The comparison takes A − B and sets it against the combined error, 1.96 × √(SE(A)² + SE(B)²), where SE is the standard error of each estimate. It is an approximate test that assumes independent random samples.

Worked example

Fictitious data taken from the guide to the post-chat survey, identical to the default values.

The Wald margins reproduce the ±5.3 and ±8.5 points in that guide. The difference is +17.0 points against a combined error of ±10.1 (z = 3.31; p < 0.001), so it exceeds sampling error. For a ±5-point margin you would need 227 responses at a CSAT of 82% and 385 in the worst case.

In NPS mode, the default values are those of the guide to NPS and sampling: 12 promoters, 4 passives and 4 detractors from 40 invitations. NPS is 40.0, with a variance of 0.6 + 0.2 − 0.4² = 0.64 and a margin of ±35.1 points (from 4.9 to 75.1). The response rate is 50%. Period B, with 13, 3 and 4, gives 45.0 ±35.3: the difference of −5.0 points falls well short of the combined error of ±49.7. For ±5 points you would need 984 responses with that distribution and 1,537 in the worst case.

How to read the result and its limits

The margin only measures random sampling error. It does not correct non-response bias: with a 12% rate, those who answer may not resemble those who do not. Nor does it correct surveys sent only after good experiences, scales that differ between periods or duplicate responses.

It does not apply the finite population correction. If you survey a large part of a small population, the real margin is somewhat smaller than the one shown.

The comparison does not work for the same people measured twice, nor for picking, after the fact, the pair of segments with the biggest gap out of many. With fewer than 30 responses, the NPS margin is only indicative. Always publish the number of responses and the response rate alongside the percentage.

Decisions it helps you make

• If a rise or fall fits within the combined error, do not present it as an improvement or use it to assess an individual.
• If the margin is too wide, extend the period or group segments before drawing conclusions.
• If you need to compare small segments, set the sample size before launching the survey.

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