NumberTally Confidence Interval Calculator
Build a confidence interval for a population mean: enter the sample mean, standard deviation, and sample size, pick 90%, 95%, or 99% confidence, and get the range instantly — with all three levels compared side by side and a visual interval plot. Free, no sign-up.
t* uses n−1 degrees of freedom.
Advanced options
The solver finds the n needed so the margin of error is at most your target, at the selected confidence level.
What a 95% confidence interval actually means
If you repeated the sampling process many times, about 95% of the intervals you compute would contain the true population mean. It does not mean there's a 95% chance the true mean is inside this particular interval — the true mean is fixed; it's the interval that would vary from sample to sample.
What makes intervals wider or narrower
Three things control the width: confidence level (99% is wider than 90% — more certainty needs more room), spread (noisier data, wider interval), and sample size (quadrupling n halves the margin, because the standard error shrinks with √n).
Frequently Asked Questions
What does a 95% confidence interval actually mean?
Across many repeated samples, 95% of the intervals built this way would capture the true population mean. It's a statement about the method's long-run reliability, not a probability about one interval.
Why does a larger sample give a narrower interval?
The margin of error is z × s/√n. Bigger n shrinks the standard error s/√n, so the interval tightens. To halve the margin you must quadruple the sample size.
When should I use 90%, 95%, or 99% confidence?
95% is the default in most fields. Use 90% when you want a tighter interval and can tolerate more risk; use 99% when the cost of missing the true value is high, accepting a wider interval.
Should I use a z-value or a t-value?
This calculator picks for you: t* (with n−1 degrees of freedom) for samples under 30, z* for 30 and up — the standard textbook rule. Override it any time with the Critical value selector. With small samples and estimated s, the t-distribution gives slightly wider, more honest intervals.
What is the margin of error?
The "±" part: margin = z × s/√n. It tells you how far the interval extends on each side of the sample mean, and it's the number pollsters quote as "±3 percentage points."