Calculate the confidence interval for a sample mean, given the standard deviation, sample size, and desired confidence level.
This tool calculates a confidence interval — a range of values likely to contain the true population parameter (such as a mean or proportion) based on sample data. Confidence intervals are widely used in research, surveys, and data analysis to express the reliability of an estimate.
Confidence Interval = Sample Mean ± (Critical Value × Standard Error). The critical value depends on the chosen confidence level (commonly 90%, 95%, or 99%) and comes from the z-distribution or t-distribution. Standard Error = Standard Deviation ÷ √(Sample Size).
Enter your sample mean, standard deviation, sample size, and desired confidence level, and the calculator returns the lower and upper bounds of the confidence interval.
Example: A survey of 100 people finds an average satisfaction score of 7.5 with a standard deviation of 1.2. At a 95% confidence level, the confidence interval is approximately 7.5 ± 0.235, or roughly (7.26, 7.73).
What does a 95% confidence interval mean? It means that if the same sampling process were repeated many times, approximately 95% of the calculated intervals would contain the true population parameter — it is not the probability that this specific interval contains the true value.
Why does a larger sample size narrow the confidence interval? Larger samples reduce the standard error, since standard error decreases as sample size increases, producing a more precise (narrower) estimate of the true population value.
Should I use a z-distribution or t-distribution? Use the t-distribution when the sample size is small (commonly under 30) or the population standard deviation is unknown; use the z-distribution for larger samples with known standard deviation.
What's the trade-off between 90%, 95%, and 99% confidence levels? Higher confidence levels produce wider intervals, offering more certainty that the interval contains the true value, but at the cost of precision.
Can confidence intervals be used for proportions, not just means? Yes, confidence intervals can be calculated for proportions (like percentages) using a similar formula adjusted for proportion data, commonly used in polling and survey analysis.