Closing the loop: sample size back into confidence interval
The sample size calculator says 385 respondents are needed for 95% confidence and a 5% margin of error, assuming a 50/50 proportion. Feeding that same 385 into the confidence interval calculator — with a sample mean of 0.5 and a standard deviation of 0.5 (the standard deviation of a proportion at exactly 50/50) — returns a margin of error of 0.049945, or 4.9945%. That's the promised 5% margin, confirmed independently by the other formula.
Why the two numbers nearly match rather than match exactly
The tiny 0.0055 percentage-point gap comes from rounding: the sample size calculator always rounds its answer up to a whole person (384.16 rounds up to 385), so the actual margin delivered by 385 real respondents is very slightly better than 5%, not slightly worse — rounding up a sample size can only tighten the guaranteed margin, never loosen it.
Why the standard deviation figure is 0.5, not something else
A single yes/no response behaves like a coin flip with probability p — its standard deviation is √(p×(1−p)), which is maximized at p=0.5, giving √0.25 = 0.5. That's exactly the "worst case" variability the sample size calculator already assumed when picking 385, so plugging it into the confidence interval calculator uses the same assumption both formulas were built around.
What this confirms about designing a survey
The two calculators solve inverse versions of the same equation: one picks a sample size to hit a target margin, the other computes the margin a given sample size actually delivers. Running a planned sample size back through the confidence interval calculator before collecting data is a useful sanity check that a survey's design will really deliver the precision it's aiming for.