MATH · CONFIDENCE INTERVAL
Confidence Interval Calculator
Calculate the confidence interval for a population mean from your sample mean, standard deviation, and sample size at 90%, 95%, or 99% confidence.
About This Calculator
A confidence interval (CI) estimates the range within which the true population mean likely falls, based on a sample. For example, a 95% CI of [94.12, 105.88] means that if you repeated the sampling process many times, about 95% of the computed intervals would contain the true mean. This calculator uses the standard-normal (z*) critical value, appropriate when the population standard deviation σ is known.
How It Works
Enter your sample mean, the population standard deviation (σ), and the sample size (n ≥ 2). Select your confidence level (90, 95, or 99%). The calculator computes the standard error (σ / √n), multiplies it by the critical z* value for your chosen level, and adds/subtracts that margin of error from the mean to produce the lower and upper bounds.
The Formula
CI = x̄ ± z* × (σ / √n)
- x̄
- sample mean
- z*
- standard normal critical value (1.96 for 95%)
- σ
- population standard deviation (must be > 0)
- n
- sample size (must be ≥ 2)
Frequently Asked Questions
- When should I use a z-interval instead of a t-interval?
- Use the z-interval (this calculator) when the population standard deviation σ is known and the sample size is reasonably large (n ≥ 30 is a common guideline). If σ is unknown and you are estimating it from the sample, use a t-interval, which accounts for the extra uncertainty in estimating σ.
- What does "95% confidence" actually mean?
- It means that if you drew many random samples and computed a CI from each, about 95% of those intervals would contain the true population mean. It does NOT mean there is a 95% probability that the true mean lies within any specific interval — once the interval is computed, the true mean either is or isn't in it.
- Why does increasing sample size narrow the confidence interval?
- The standard error SE = σ / √n decreases as n grows, because larger samples give a more precise estimate of the mean. A narrower CI reflects that precision.
- What are the standard critical z* values?
- 90% confidence: z* = 1.6449. 95% confidence: z* = 1.9600. 99% confidence: z* = 2.5758. These are the z-scores such that Φ(z*) = (1 + confidence_level)/2 in the standard normal distribution.