Confidence Interval Calculator
Estimate the range where a population mean likely falls using your sample data. Use this when reporting survey results, lab measurements, or any study where you need to quantify uncertainty around an estimate.
About this calculator
A confidence interval gives a range of plausible values for a population mean based on sample data. The margin of error is calculated as E = z* × (σ / √n), where z* is the critical value for your chosen confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), σ is the standard deviation, and n is the sample size. The full interval is then [x̄ − E, x̄ + E], where x̄ is the sample mean. A 95% confidence interval means that if you repeated the sampling process many times, approximately 95% of the resulting intervals would contain the true population mean. Wider intervals reflect more uncertainty — caused by smaller samples or higher variability in the data.
How to use
Suppose a researcher measures the resting heart rate of 36 adults, finding a sample mean of 72 bpm and a standard deviation of 12 bpm, and wants a 95% confidence interval. Step 1: Identify z* = 1.96 for 95%. Step 2: Compute SE = 12 / √36 = 12 / 6 = 2. Step 3: Margin of error E = 1.96 × 2 = 3.92. Step 4: Interval = [72 − 3.92, 72 + 3.92] = [68.08, 75.92]. You can be 95% confident the true mean resting heart rate lies between 68.08 and 75.92 bpm.
Frequently asked questions
What does a 95% confidence interval actually mean in statistics?
A 95% confidence interval means that if you drew many random samples from the same population and computed an interval each time, about 95% of those intervals would contain the true population mean. It does not mean there is a 95% probability that the true mean falls in your specific interval — once calculated, the interval either contains the true mean or it does not. The 95% refers to the reliability of the method, not a single outcome. This distinction is important for correctly interpreting study results.
How does sample size affect the width of a confidence interval?
Larger sample sizes produce narrower confidence intervals, reflecting greater precision in your estimate. Because sample size n appears under a square root in the formula SE = σ / √n, you need to quadruple your sample size to halve the margin of error. For example, increasing n from 25 to 100 cuts the standard error in half. This is why large studies are generally preferred — they yield tighter, more informative intervals without changing the confidence level.
When should I use a 90% versus 99% confidence interval?
The choice of confidence level depends on the consequences of being wrong and the required precision. A 90% interval is narrower and suitable when a slightly higher error risk is acceptable, such as in exploratory research or business forecasting. A 99% interval is wider but offers greater assurance, making it appropriate for medical trials, safety testing, or policy decisions where mistakes are costly. A 95% confidence level is the most widely used default across scientific disciplines as a balanced compromise between precision and reliability.