Confidence intervals for small-team scores, and why the t-distribution is the honest choice

By Zak Fenton, MSc Workplace Health & Wellbeing, Alltoogether · Published 12 July 2026 · Last reviewed 12 July 2026 · Every claim is sourced; report errors and we fix them visibly.

Why a small group needs a wider interval

A confidence interval on a mean has two sources of uncertainty: how much individuals vary, and how well the sample estimates that variation. With a large sample, the second source is negligible and the normal distribution is a fine approximation. With a small sample, the sample standard deviation is itself uncertain, and ignoring that produces an interval that is too narrow, which reads as false precision. William Gosset worked this out for exactly the small-sample case in 1908 (publishing as "Student"), and the t-distribution is the result: heavier tails than the normal, converging on it as the group grows.

The practical effect is largest where it matters most for workplace wellbeing, in small teams. For a group of 5, the 95% t-multiplier is 2.776 against the normal's 1.96, roughly 40% wider. By a group of 30 the gap has nearly closed (2.045 versus 1.96). Using z on a team of 6 does not make the team's score more certain; it just prints a narrower range than the data can support.

The method, verbatim

For a group of n respondents with sample mean m and sample standard deviation s, the 95% confidence interval is:

half_width = t(0.975, n-1) * s / sqrt(n)
ci_low  = m - half_width
ci_high = m + half_width

The engine stores t(0.975, n-1) in a lookup function (iwe.t975) so every deployment computes the identical multiplier, and the interval is written to the aggregate row alongside the value. It is not recomputed in the dashboard, where a client could quietly swap in a narrower method. The interval is a stored fact, checked at certification.

Try it: the 95% t-interval on a small-group mean

How this pairs with the n≥5 floor

The floor and the interval solve different halves of the same honesty problem. The n≥5 floor decides whether a group is large enough to show a value at all, without re-identifying anyone. The confidence interval, for groups above the floor, tells you how much to trust the value it shows. Below five, there is no value and no interval, only a coded suppression reason. From five to nine, there is a value, a wide interval, and a provisional label. The full set of constraints is in the honesty rules.

What to ask any vendor about their intervals

Do your team scores carry a confidence interval at all, or just a number? Is it computed with the t-distribution or the normal, and for a team of 6 what multiplier do you use? Is the interval stored with the value or recomputed at display time? Does a small-group score carry a provisional label, and does that label travel into exports? A vendor who cannot answer the second question is probably using z, and printing more certainty than the group size earns.

See it running

The interactive demo shows the same t-method firing live on a simulated team as the group shrinks. Free for any employer at alltoogether.com · the method at openworkplacehealth.org · building it into your product is what this site is for.