What a confidence interval really says
A confidence interval turns a single point estimate into a range that reflects sampling uncertainty. The interval is centred on your sample mean and extends a margin of error in each direction. That margin is the critical Z value for your chosen confidence level multiplied by the standard error, which is the standard deviation divided by the square root of the sample size.
The confidence level is best read as a property of the method, not of one interval. Saying "95% confident" means that if you repeated the whole sampling process many times and built an interval each time, about 95% of those intervals would contain the true population mean. Any single interval either contains it or does not.
What makes an interval narrow or wide
The chart shows the lower bound, the sample mean and the upper bound so you can see the spread at a glance. Three levers control how wide that spread is:
- Sample size: the margin shrinks with the square root of n, so quadrupling the sample roughly halves the width.
- Variability: a larger standard deviation means a wider interval, because the data is noisier.
- Confidence level: demanding 99% rather than 90% confidence uses a bigger Z value and widens the interval.
Assumptions and limits
This calculator uses Z values from the normal distribution, which is appropriate when the sample is reasonably large (a common rule of thumb is n of 30 or more) or when the population standard deviation is known. For small samples with an unknown population standard deviation, a t-distribution gives wider, more honest intervals.
A confidence interval also assumes your data is a random, representative sample. It accounts for random sampling error only — it cannot fix bias from a flawed sampling method, so a tight interval around a biased estimate is still misleading.
Formula
interval = mean ± Z·(stdDev / √n)Frequently asked questions
- What is the margin of error?
- The margin of error is the half-width of the interval, Z·(stdDev/√n). The interval runs from the mean minus this margin to the mean plus it.