How does a confidence interval change with sample size and variance?

Why

A confidence interval expresses uncertainty around an estimate. For an independent sample mean, a familiar approximation is mean plus or minus a critical value multiplied by s divided by the square root of n. This reveals two mechanisms.

Increasing the number of independent observations reduces sampling uncertainty, whereas greater variation across observations increases it. The critical value also depends on confidence level and sample size. For example, if a study samples four times as many independent donors with similar variation, its standard error is approximately halved, not quartered.

However, recording four measurements from the same donor is not four independent biological samples. Dependence, unequal variance, skew, small samples and complex designs can require different interval methods. Most importantly, no amount of precision eliminates selection bias, measurement bias or a confounded design.

Report the estimate, interval, unit of independence and assumptions rather than promising that a narrower interval means a better study.