Confidence interval calculator
Margin of error scales with one over the square root of the sample size, so cutting it in half takes four times the data. That is why polls hover around a thousand respondents: it buys roughly ±3 points, and getting to ±1.5 would need four thousand at four times the cost. The relationship is also why a sample of 1,000 is nearly as good for a country of 60 million as for a town of 60,000 — population size barely enters into it.
Uses the normal approximation. For samples under about 30, or an unknown population standard deviation, a t-distribution gives a wider and more honest interval.
A confidence interval is the mean plus and minus the critical value times the standard error. A mean of 100 with a standard deviation of 15 and n = 100 gives a 95% interval of 97.06 to 102.94 — a margin of error of ±2.94.
How to calculate a confidence interval
The interpretation is narrower than it sounds. A 95% confidence interval does not mean there is a 95% chance the true value lies inside this particular interval — the true value is fixed and either is or is not in it. It means that if you repeated the whole sampling process many times, 95% of the intervals produced would contain the true value. It is a statement about the method, not about the one interval in front of you.
Questions
That 95% of intervals built this way from repeated samples would contain the true value. It is about the method, not this interval.