Confidence Interval Calculator

Frequently Asked Questions

Why does a larger sample give a narrower confidence interval?

Because SE = SD ÷ √n decreases as n increases. Larger samples yield more precise estimates of the mean. To halve the margin of error you need to quadruple the sample size.

What if I do not know the standard deviation?

In that case, use the t-distribution and enter the sample standard deviation s (computed with n−1 in the denominator). The t-distribution accounts for the additional uncertainty in estimating σ from the data.

Why is the 99% CI wider than the 95% CI?

Higher confidence requires a wider net. The 99% critical value (2.576) is larger than the 95% one (1.96), producing a wider interval. You pay for more confidence with less precision.

Can I use this for proportions?

This calculator is for a population mean, not a proportion. For proportions, use the Proportion Confidence Interval calculator, which uses p̂(1−p̂) as the variance estimate instead of s².

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Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

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How This Calculator Works

A confidence interval for the population mean is built from the sample mean, the standard deviation, and the sample size. The formula is:

CI = x̄ ± critical × SE

where SE = SD ÷ √n is the standard error of the mean. The critical value comes from the z-distribution (for large n or known population σ) or the t-distribution with n−1 degrees of freedom (for small n or unknown σ). This calculator supports confidence levels of 90%, 95%, 99%, and 99.9% with the corresponding z-critical values of 1.645, 1.96, 2.576, and 3.291.

Z vs. T: Which to Choose

Use the z-distribution when the population standard deviation σ is known, or when the sample size n ≥ 30 (by the Central Limit Theorem, the sample mean is approximately normal regardless of the population shape).

Use the t-distribution when the population σ is unknown and must be estimated from the sample - which is almost always the case in practice. The t-distribution has heavier tails than the normal, producing a wider interval to compensate for the additional uncertainty in estimating σ. As n grows, t converges to z; for n ≥ 100 the difference is negligible.

Worked Example

A sample of n = 100 has mean x̄ = 50 and SD = 10. At 95% confidence using z: SE = 10 ÷ √100 = 1.0. Margin of error = 1.96 × 1.0 = 1.96. The interval is (50 − 1.96, 50 + 1.96) = (48.04, 51.96).

If n were only 25 with the same mean and SD using t (df = 24, critical ≈ 2.064): SE = 10 ÷ √25 = 2.0. Margin = 2.064 × 2.0 = 4.128. Interval: (45.87, 54.13) - wider because of smaller n and the heavier-tailed t.

Correct Interpretation of a Confidence Interval

A 95% confidence interval does not mean there is a 95% probability that the true mean lies inside this particular interval. The true mean is a fixed (unknown) constant, not a random variable. The interval is what varies from sample to sample.

The correct interpretation: if you repeated the entire sampling procedure many times, about 95% of the intervals constructed this way would contain the true population mean. This specific interval either contains the true mean or it does not - we just do not know which.

Common Mistakes

Reading a CI as a probability statement about a fixed parameter rather than about the procedure.

Using z when n < 30 and σ is unknown - the t-distribution is more appropriate.

Entering variance where standard deviation is expected, inflating the interval by √σ.

Forgetting that halving the margin of error requires quadrupling the sample size, because SE scales with 1/√n.

Assuming a CI that does not include zero proves the null is false - you need a formal hypothesis test for that claim.

Confidence Interval Calculator

Frequently Asked Questions

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