This calculator helps you estimate the true population mean with a specified level of confidence, providing both precision and uncertainty measures for your data. It automatically determines whether to use z-test or t-test methods, handles both raw data uploads and manual statistic entry, and provides comprehensive results including confidence intervals, margins of error, critical values, and detailed statistical interpretation.
A confidence interval for a mean is a statistical range that estimates where the true population mean likely falls, based on sample data. It consists of a lower bound and upper bound, calculated using the sample mean plus and minus a margin of error.
Unlike a point estimate (which gives just one number), a confidence interval acknowledges the uncertainty inherent in sampling and provides a range of plausible values along with our level of confidence in that range.
Simple Example: Imagine you want to know the average height of all students at a university. You measure 50 randomly selected students and find their average height is 68.5 inches.
Instead of saying "the average height is exactly 68.5 inches," a 95% confidence interval might give you [67.2, 69.8] inches.
This means: "Based on our sample, we're 95% confident the true average height of all students at this university is between 67.2 and 69.8 inches."
Want to see how confidence intervals behave with different sample sizes and confidence levels?
Try our Interactive Confidence Interval SimulationThe formula depends on whether the population standard deviation is known:
The margin of error is the value that gets added and subtracted from the sample mean to create the confidence interval bounds:
For Z-Test:
For T-Test:
The confidence level (commonly 90%, 95%, or 99%) represents how confident we are in our interval estimation method, not in any single interval.
Narrower interval, less certainty. Good for preliminary analysis.
Standard choice. Balances precision with confidence.
Wider interval, higher certainty. Used for critical decisions.
Important: If we repeated our sampling process 100 times and calculated 95% confidence intervals each time, approximately 95 of those intervals would contain the true population mean.
Uses standard normal distribution
Uses t-distribution with n-1 degrees of freedom
Note: As sample size increases, the t-distribution approaches the standard normal distribution, so the choice becomes less critical for large samples.
"We are 95% confident that the true population mean lies between [lower bound] and [upper bound]."
❌ "There's a 95% probability the population mean is in this interval."
❌ "95% of the data falls within this interval."
Watch how confidence intervals behave across multiple samples. Each line represents a confidence interval from a different sample.
Key insight: About 95% of the confidence intervals should capture the true mean. As you run more samples, the capture rate should approach 95%.
This calculator helps you estimate the true population mean with a specified level of confidence, providing both precision and uncertainty measures for your data. It automatically determines whether to use z-test or t-test methods, handles both raw data uploads and manual statistic entry, and provides comprehensive results including confidence intervals, margins of error, critical values, and detailed statistical interpretation.
A confidence interval for a mean is a statistical range that estimates where the true population mean likely falls, based on sample data. It consists of a lower bound and upper bound, calculated using the sample mean plus and minus a margin of error.
Unlike a point estimate (which gives just one number), a confidence interval acknowledges the uncertainty inherent in sampling and provides a range of plausible values along with our level of confidence in that range.
Simple Example: Imagine you want to know the average height of all students at a university. You measure 50 randomly selected students and find their average height is 68.5 inches.
Instead of saying "the average height is exactly 68.5 inches," a 95% confidence interval might give you [67.2, 69.8] inches.
This means: "Based on our sample, we're 95% confident the true average height of all students at this university is between 67.2 and 69.8 inches."
Want to see how confidence intervals behave with different sample sizes and confidence levels?
Try our Interactive Confidence Interval SimulationThe formula depends on whether the population standard deviation is known:
The margin of error is the value that gets added and subtracted from the sample mean to create the confidence interval bounds:
For Z-Test:
For T-Test:
The confidence level (commonly 90%, 95%, or 99%) represents how confident we are in our interval estimation method, not in any single interval.
Narrower interval, less certainty. Good for preliminary analysis.
Standard choice. Balances precision with confidence.
Wider interval, higher certainty. Used for critical decisions.
Important: If we repeated our sampling process 100 times and calculated 95% confidence intervals each time, approximately 95 of those intervals would contain the true population mean.
Uses standard normal distribution
Uses t-distribution with n-1 degrees of freedom
Note: As sample size increases, the t-distribution approaches the standard normal distribution, so the choice becomes less critical for large samples.
"We are 95% confident that the true population mean lies between [lower bound] and [upper bound]."
❌ "There's a 95% probability the population mean is in this interval."
❌ "95% of the data falls within this interval."
Watch how confidence intervals behave across multiple samples. Each line represents a confidence interval from a different sample.
Key insight: About 95% of the confidence intervals should capture the true mean. As you run more samples, the capture rate should approach 95%.
