Confidence Interval Calculator

Last reviewed: June 2026 by the InvNorm Calculator Editorial Team. Report an issue

Use this Confidence Interval Calculator to construct a confidence interval for a population mean using the Z-based method. Enter the sample mean, standard deviation (or standard error), sample size, and your desired confidence level. The calculator will compute the margin of error, critical Z value, and the lower and upper bounds of the interval.

What Is a Confidence Interval?

A confidence interval provides a range of plausible values for a population parameter based on sample data. When you compute a 95% confidence interval for a population mean, you are constructing an interval such that if you repeated the sampling process many times, approximately 95% of the resulting intervals would contain the true population mean. The confidence level reflects how often the method produces an interval that captures the parameter, not the probability that any single interval contains it.

Confidence intervals are more informative than point estimates alone because they convey the precision of the estimate. A narrow interval suggests high precision (typically from a large sample or low variability), while a wide interval indicates substantial uncertainty. Researchers, analysts, and decision-makers use confidence intervals to quantify uncertainty in estimates of means, proportions, differences, and other parameters.

Confidence Interval Formula

The Z-based confidence interval for a population mean is:

CI = x̄ ± z* · SE

where:

• x̄ is the sample mean
• z* is the critical Z value for the chosen confidence level, computed as z* = Φ⁻¹(1 − α/2)
• SE is the standard error of the mean, calculated as SE = σ / √n
• α = 1 − confidence level (e.g., α = 0.05 for 95% confidence)

The margin of error (ME) is the quantity z* · SE. The lower bound is x̄ − ME and the upper bound is x̄ + ME.

Worked Example

A researcher measures the systolic blood pressure of 64 patients and finds a sample mean of 120 mmHg with a known population standard deviation of 16 mmHg. Construct a 95% confidence interval.

1. Identify values: x̄ = 120, σ = 16, n = 64, confidence level = 95%
2. Find the critical value: α = 0.05, z* = Φ⁻¹(0.975) ≈ 1.95996
3. Calculate standard error: SE = 16 / √64 = 16 / 8 = 2.0
4. Calculate margin of error: ME = 1.95996 × 2.0 = 3.920
5. Construct the interval: 120 ± 3.920 = (116.080, 123.920)
6. Interpret: We are 95% confident that the true mean systolic blood pressure lies between 116.08 and 123.92 mmHg.

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