Confidence Interval Calculator
Calculate a confidence interval for a sample mean at 80–99% confidence levels.
Cómo funciona
A confidence interval gives a range of plausible values for a true population mean, based on a sample — rather than reporting a single number as if it were exact, it acknowledges the uncertainty inherent in estimating from a sample rather than measuring an entire population.
The calculation starts with the standard error: sample standard deviation divided by the square root of the sample size. This shrinks as sample size grows, reflecting that larger samples give more precise estimates of the true population mean.
The margin of error is then z-score × standard error, where the z-score corresponds to your chosen confidence level (1.96 for 95% confidence is the most commonly cited value). The confidence interval is the sample mean plus and minus that margin of error.
For example, a sample mean of 100 with a standard deviation of 15 and a sample size of 30 has a standard error of 15 ÷ √30 ≈ 2.74. At 95% confidence (z=1.96), the margin of error is about 5.37, giving a confidence interval of roughly 94.6 to 105.4 — meaning we're 95% confident the true population mean falls somewhere in that range.
Choosing a higher confidence level (98% or 99% instead of 95%) widens the interval, since being more certain about capturing the true value requires a broader range — there's a direct tradeoff between confidence and precision that no formula can escape.
A confidence interval is often more useful than a single-point estimate alone because it directly communicates the uncertainty in the estimate — two studies could report the identical sample mean, but one with a much larger sample (and therefore a narrower confidence interval) represents meaningfully stronger evidence than the other.
This calculator uses the z-score (normal distribution) method, which assumes a reasonably large sample size or a known population standard deviation. For small samples (typically under 30) with an estimated standard deviation, statisticians often prefer the t-distribution instead, which produces a slightly wider, more conservative interval to account for the added uncertainty of estimating variability from limited data.
Which confidence level should I use?
95% is the most common default in research and reporting; higher levels (98%, 99%) give a wider, more conservative interval.
What does a 95% confidence interval actually mean?
If you repeated the sampling process many times, about 95% of the resulting confidence intervals would contain the true population mean — it's a statement about the reliability of the method, not a 95% probability about this specific interval.
Why does a larger sample size narrow the confidence interval?
Standard error decreases as sample size increases (dividing by a larger square root), which directly shrinks the margin of error and therefore the width of the interval — larger samples give more precise estimates.
Why does a higher confidence level widen the interval?
Greater certainty requires a wider range to be confident of capturing the true value — there's an inherent tradeoff between how confident you want to be and how precise (narrow) the resulting interval can be.
What z-scores correspond to common confidence levels?
Approximately 1.28 for 80%, 1.44 for 85%, 1.645 for 90%, 1.96 for 95%, 2.33 for 98%, and 2.576 for 99% confidence.
Do I need to know the population standard deviation, or is a sample estimate okay?
This calculator uses the standard deviation value you provide, whether from the full population or estimated from your sample — for smaller samples with an estimated standard deviation, a t-distribution technically gives a more precise interval than the z-score method used here.
Why report a confidence interval instead of just the mean?
It communicates the uncertainty in the estimate — a mean alone looks precise even when based on limited data, while the interval width honestly reflects how much confidence the data actually supports.
Can a confidence interval be used for anything other than a mean?
Yes, conceptually — confidence intervals can be constructed for proportions, differences between groups, and many other statistics, though each uses a different underlying formula than the one this calculator applies to sample means.
When should I use a t-distribution instead of the z-score method?
When your sample size is small (commonly under 30) and you're estimating the standard deviation from the sample itself rather than knowing the true population value — the t-distribution accounts for this added uncertainty with a wider interval.
Does a wider confidence interval mean the estimate is less accurate?
Not necessarily less accurate, but less precise — it reflects the actual amount of uncertainty given the sample size and variability, which is more honest than an artificially narrow range.