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Confidence Interval Calculator

Confidence interval for a mean from sample statistics.

What this calculator does

This tool computes a confidence interval around a sample mean—the range where you can reasonably expect the true population mean to lie. You input your sample statistics and choose a confidence level, and it calculates both the interval bounds and the margin of error.

The formula

CI = x̄ ± (t* × SE) where SE = SD / √n

Here, x̄ is your sample mean, t* is the critical value from the t-distribution (or z-distribution for very large samples), SD is the standard deviation, and n is the sample size. The margin of error is the ± part.

Worked example

Imagine you surveyed 50 gym members about their weekly exercise hours. Your results:

  • Sample mean: 4.2 hours
  • Standard deviation: 1.8 hours
  • Sample size: 50
  • Confidence level: 95%

Step 1: Calculate standard error

SE = 1.8 ÷ √50 = 1.8 ÷ 7.07 = 0.255 hours

Step 2: Find the critical value

With n = 50 and 95% confidence, the t-critical value is approximately 2.01 (from t-distribution tables, using 49 degrees of freedom).

Step 3: Calculate margin of error

Margin of error = 2.01 × 0.255 = 0.512 hours

Step 4: Build the interval

Lower bound = 4.2 − 0.512 = 3.688 hours

Upper bound = 4.2 + 0.512 = 4.712 hours

Result: You can say with 95% confidence that the true average weekly exercise time for all gym members falls between 3.69 and 4.71 hours.

If you'd chosen 99% confidence instead, the critical value would be about 2.68, pushing the margin of error to 0.68 hours and widening your interval to roughly 3.52–4.88 hours.

Common mistakes

Confusing the sample SD with the standard error: The standard deviation describes variation in your data; the standard error (which is smaller) describes precision of your mean estimate. Always divide SD by √n to get SE.

Misinterpreting the interval as fixed: The interval itself doesn't have a 95% probability of containing the true mean—it either does or doesn't. The 95% refers to the long-run success rate of the method.

Forgetting that sample size matters enormously: Doubling your sample size reduces SE by a factor of √2 (about 1.41), narrowing your interval significantly. This is often more practical than demanding higher confidence.

Using z instead of t for small samples: For n < 30, always use the t-distribution. It has heavier tails and produces appropriately wider intervals when data is limited.

This calculator provides a statistical estimate, not professional advice. For critical decisions involving confidence intervals, consult a statistician.

Frequently asked questions

What does a 95% confidence interval actually mean?

It means if you repeated your sampling process 100 times, approximately 95 of those times the true population mean would fall within the calculated range. It's not a guarantee for any single interval, but a long-run probability statement.

Why does a higher confidence level produce a wider interval?

To be more confident you've captured the true mean, you need a wider net. A 99% confidence interval is wider than a 95% interval from the same sample because you're asking for greater certainty.

What's the difference between standard deviation and standard error?

Standard deviation measures spread in your actual sample data. Standard error (SD ÷ √n) measures how much your sample mean varies from the true population mean. Larger samples have smaller standard errors.

Can I use this with non-normal data?

For large samples (n > 30), the Central Limit Theorem makes this method robust even if data isn't perfectly normal. For smaller samples from non-normal populations, consider consulting a statistician.

What if my sample size is very small?

With n < 30, this calculator uses the t-distribution instead of the normal distribution, which automatically widens the interval to account for extra uncertainty from limited data.

How do I reduce the margin of error?

Increase your sample size (larger n shrinks the interval) or accept lower confidence (95% instead of 99%). You cannot reduce it by changing the standard deviation—that's a property of your data.

Related calculators

Sample Size CalculatorRequired survey sample size for a confidence level and margin of error.Standard Deviation CalculatorStandard deviation and variance for a population or a sample.Z-Score CalculatorStandardize a value and find its percentile under the normal curve.P-Value CalculatorConvert a z-score into one- and two-tailed p-values.