Confidence Interval Calculators

confidence intervals

CI for Mean (σ known)

Confidence interval for a population mean using z-distribution.

Example answer(48.6293, 51.3707)

This is the worked example that ships with the calculator. Replace the values above with your own and press Calculate.

Sample Mean (x̄): 50
Population Std Dev (σ): 5
Sample Size (n): 36
Confidence Level: 90%

90% Confidence Interval (z-based)FormulaCI = x̄ ± z* · (σ / √n)
x̄ = 50.0000, σ = 5.0000, n = 36
Critical value z*Substituteα = 0.1000, z* = z_{0.9500}
1.6449
Standard errorSubstituteSE = 5.0000 / √36
0.8333
Margin of errorSubstituteE = 1.6449 · 0.8333
1.3707
Confidence intervalSubstitute(50.0000 - 1.3707, 50.0000 + 1.3707)
(48.6293, 51.3707)

Free reference

Confidence Intervals: the formulas

A confidence interval turns a single sample estimate into a range of plausible values for the population parameter: the estimate plus or minus a critical value times the standard error.

Confidence Interval for a Mean: xbar +/- critical value * standard error - Estimate plus or minus a margin of error.
Standard Error of xbar: SE_xbar = sigma / sqrt(n) - Standard deviation of the sampling distribution of the sample mean.

See the full formula reference

Worked example: CI for a Mean (sigma known)

A sample of 25 measurements has mean 50, and the population standard deviation is known to be 5. Build a 95% confidence interval for the population mean.

Inputs. Sample mean 50, sigma = 5, n = 25, confidence level 95%.

95% Confidence Interval (z-based). CI = x̄ ± z* · (σ / √n) gives x̄ = 50.0000, σ = 5.0000, n = 25
Critical value z*. α = 0.0500, z* = z_{0.9750} gives 1.9600
Standard error. SE = 5.0000 / √25 gives 1.0000
Margin of error. E = 1.9600 · 1.0000 gives 1.9600
Confidence interval. (50.0000 - 1.9600, 50.0000 + 1.9600) gives (48.0400, 51.9600)

Answer: (48.04, 51.96)

The standard error is 5 / sqrt(25) = 1, the margin of error is 1.96 * 1 = 1.96, and the interval runs from 48.04 to 51.96.

Frequently asked questions

What does 95% confidence actually mean?

It describes the procedure, not one interval: if you drew many samples and built an interval from each, about 95% of those intervals would contain the true population mean. Any single interval, like (48.04, 51.96) above, either contains it or does not.

What makes a confidence interval narrower?

A larger sample size, a smaller standard deviation, or a lower confidence level. The margin of error is the critical value times sigma / sqrt(n), so quadrupling the sample size halves the width at the same confidence level.

When do I use z versus t for a confidence interval?

Use the z critical value when the population standard deviation is known, as in the worked example. When sigma is estimated from the sample, use the t critical value with n - 1 degrees of freedom, which widens the interval to reflect the extra uncertainty.