Confidence Interval Calculator

In short A confidence interval is the sample mean plus and minus a margin of error, where that margin is a critical value times the standard error. With a mean of 100, a standard deviation of 15 and 25 observations, the standard error is 3.00, the 95% critical t on 24 degrees of freedom is 2.0639, and the interval runs from 93.81 to 106.19. Use t rather than z unless the standard deviation came from outside the sample — a calibrated instrument or a published population figure — because a standard deviation estimated from the data carries its own error, and t is what accounts for it. The penalty is largest exactly where people are most tempted to ignore it: at n = 5 the t interval is 41.66% wider than the z interval, at n = 30 it is 4.35% wider, and by n = 100 it is 1.24%. The single most useful fact here is how the interval shrinks. The margin falls with the square root of the sample size, so halving it takes four times the data, not twice. And the interpretation is narrower than it looks. A 95% interval does not mean there is a 95% chance the true mean lies inside this one. The true mean is a fixed number; it either is or is not in this interval. The 95% is the long-run hit rate of the method, which is a statement about the procedure rather than about your particular answer.