Confidence Interval Calculator
Calculate confidence intervals and margin of error for sample means. Perfect for statistics, research, and data analysis.
Input Data
Visualization
The confidence interval represents the range where the true population mean is likely to be found.
Calculation Results
Understanding Confidence Intervals
What is a Confidence Interval?
A confidence interval is a range of values that likely contains the true population parameter. It’s calculated from sample data and provides a measure of uncertainty.
A 95% confidence interval means that if we repeated the sampling process 100 times, 95 of the confidence intervals would contain the true population mean.
How to Calculate
The confidence interval for a mean is calculated using the formula:
Where:
- x̄ is the sample mean
- Z is the Z-score for the confidence level
- σ is the population standard deviation
- n is the sample size
Interpretation
When you say “We are 95% confident that the true mean is between X and Y”, it means:
- 95% of confidence intervals from repeated sampling will contain the true mean
- There’s a 5% chance that the interval does not contain the true mean
- The interval gives a range of plausible values for the population mean
Common Z-Scores
90% Confidence Level
Z-Score: 1.645
Use when you need a narrower interval but can accept more uncertainty. Common in exploratory research.
95% Confidence Level
Z-Score: 1.96
The most commonly used confidence level. Provides a good balance between precision and reliability.
99% Confidence Level
Z-Score: 2.576
Use when you need high certainty. Produces wider intervals but greater confidence in results.