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Confidence Interval Calculator

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Enter your sample mean, standard deviation, and sample size to find the confidence interval for your data.

Confidence Interval Calculator

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Confidence interval
72.23 to 77.77
Margin of error
±2.77
Standard error
1.4142
Uses the z-distribution, appropriate for large samples (n ≥ 30). Smaller samples are more accurately handled with a t-distribution.

The formula

Standard error = Standard deviation / sqrt(Sample size) Margin of error = z* x Standard error Confidence interval = Sample mean ± Margin of error
Example

A sample mean of 75, standard deviation of 10, sample size of 50, at 95% confidence: margin of error ≈ 2.77, giving a confidence interval of 72.23 to 77.77.

Step-by-step guide

  1. Enter your sample mean and standard deviation.
  2. Enter your sample size.
  3. Choose a confidence level - 95% is the most commonly used default.
  4. Read the confidence interval - the range likely to contain the true population mean.

What "95% confidence" actually means

A 95% confidence interval doesn't mean there's a 95% chance the true population mean falls within this specific interval - the true mean either is or isn't in there, with no probability about it once the data is collected. What it actually means: if you repeated this same sampling process many times and built a new interval each time, about 95% of those intervals would contain the true population mean. It's a statement about the reliability of the method, not a probability about this one specific result.

Common mistakes

Interpreting a 95% confidence interval as "95% probability the true mean is in this range" - the correct interpretation is about the long-run reliability of the method, not a probability statement about this specific interval.
Using the z-distribution for a small sample size (typically under 30) - a t-distribution is more accurate in that case, since it accounts for the added uncertainty of estimating standard deviation from a small sample.

Frequently asked questions

Why does a higher confidence level give a wider interval?

Being more confident that an interval captures the true value requires casting a wider net - a 99% confidence interval is necessarily wider than a 90% one for the same data, trading precision for greater certainty.

Why does a larger sample size narrow the interval?

A larger sample gives a more precise estimate of the true population mean, which shrinks the standard error (since it's divided by the square root of sample size) and therefore narrows the resulting confidence interval.

What if my sample size is small?

For samples under about 30, a t-distribution (which has slightly wider critical values than the z-distribution used here) gives a more accurate confidence interval, accounting for the extra uncertainty in a smaller dataset.

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