How to Calculate a Mean Confidence Interval
A confidence interval combines an estimate, its standard error and a critical value. Lynara labels this tool as a normal/z approximation so the method is explicit.
Method
- Enter the sample or estimated mean.
- Enter the standard deviation and sample size.
- Choose a 90%, 95% or 99% confidence level.
- Multiply standard error by the z critical value and add/subtract the margin from the mean.
Worked example
Mean 50, SD 10 and n = 100 gives SE = 1.
At 95%, z* ≈ 1.96, so the interval is about 48.04 to 51.96.
Notes
- Small samples with unknown population SD may require a Student t interval.
- Confidence level changes the critical value and interval width.
- Follow the statistical method required by your course or analysis.
- A confidence interval calculation also depends on the critical-value model. Lynara's stated normal-critical-value method should not be silently substituted for a t-based interval when the course or analysis requires a t distribution, especially with small samples and estimated population variability.
Worked sampling case: when the population standard deviation is known
Assume a simple random sample of 64 independent observations has mean 72 and that the population standard deviation is known to be 12. These are illustrative inputs, not a real study. The standard error is 12 ÷ square root of 64 = 1.5. Under an appropriate normal-model assumption, a two-sided 95% z interval uses 1.96 as the illustrative critical value, making the margin of error 1.96 × 1.5 = 2.94. The interval is 72 − 2.94 to 72 + 2.94, or 69.06 to 74.94.
Changing confidence while holding the same observations and known population standard deviation changes interval width. For example, with a 99% critical z value approximated as 2.576, the margin would be 3.864 and the interval approximately 68.14 to 75.86. The higher confidence procedure makes a wider interval; it does not imply that the sample mean itself changed. Rounding the endpoints prematurely can make neighboring results look inconsistent.
If the population standard deviation is not actually known, do not type the sample standard deviation into a field that expects a known population sigma and describe the result as the same exact interval. A t-based interval with appropriate degrees of freedom may be required. Also check sampling independence, outliers, selection bias and whether the sampling distribution is adequately approximated by the model. A narrow interval from a biased sample can be precise around the wrong target.
- Known population sigma: 12; sample size: 64; sample mean: 72.
- Standard error: 1.5; 95% normal margin: 2.94; z interval: 69.06–74.94.
- Check whether the tool assumes a z interval or a t interval before copying the result into a report.
A confidence interval combines an estimate with sampling uncertainty
A basic interval takes the sample mean and adds and subtracts a critical value multiplied by the standard error. The width depends on variability, sample size and the chosen confidence level. Higher confidence uses a larger critical value and produces a wider interval.
This guide's normal-approximation method is most appropriate when a z-based approach is justified. When the population standard deviation is unknown and sample size is limited, a t interval is often more appropriate.
Interpret confidence without turning it into a probability about one fixed interval
A 95% confidence procedure is designed so that, under its assumptions, intervals from repeated samples cover the true parameter about 95% of the time. After one interval is calculated, the parameter is not treated as randomly moving inside and outside that fixed range in the standard frequentist interpretation.
Report the estimate and interval together. The interval shows precision; it does not automatically establish practical importance or correct problems in the sampling design.
Authoritative references
Primary or authoritative references for the external standard, scoring rule or identifier convention used on this page.
Editorial review and limitations
Continue with the related tools
Use the linked tools for your own inputs and scenarios.
- Standard Error Calculator — Calculate the standard error of a sample mean from standard deviation and sample size.
- Z-Score & Normal Percentile Calculator — Calculate a z-score from value, mean and standard deviation, plus normal-model percentile and right-tail probability.
- Margin of Error Calculator — Calculate a normal-approximation margin of error for a proportion from sample size, estimated proportion and confidence level.