Confidence Interval Calculator
Calculate confidence intervals for population means or proportions. Choose your confidence level (90%, 95%, or 99%) and enter your sample statistics to get the interval range and margin of error.
How to Use This Confidence Interval Calculator
This calculator computes confidence intervals for both population means and proportions. Select the appropriate calculation type using the tabs and follow these steps:
For Confidence Interval of a Mean:
- Enter your sample mean (x-bar), which is the average of your sample data
- Enter the standard deviation of your sample or population
- Enter your sample size (number of observations)
- Select your desired confidence level (90%, 95%, or 99%)
- Click Calculate to see the confidence interval range and margin of error
For Confidence Interval of a Proportion:
- Enter the number of successes (favorable outcomes) in your sample
- Enter your total sample size
- Select your desired confidence level
- Click Calculate to see the interval range for the true population proportion
The visual representation shows your sample statistic as a point within the confidence interval range, helping you understand the uncertainty in your estimate.
What is a Confidence Interval?
A confidence interval is a range of values constructed from sample data that is likely to contain the true population parameter. Unlike a single point estimate, a confidence interval acknowledges that samples vary and provides a range that quantifies this uncertainty.
The confidence level (such as 95%) tells you how reliable the interval construction method is. A 95% confidence interval means that if you repeated your study many times with new samples, approximately 95% of the calculated intervals would contain the true population value. It does not mean there is a 95% probability that this specific interval contains the true value (the true value either is or is not in the interval).
Confidence intervals are preferred over point estimates in research and decision-making because they communicate the precision of your estimate. A narrow interval indicates a precise estimate, while a wide interval suggests more uncertainty. Factors affecting interval width include sample size (larger samples produce narrower intervals), variability in the data (more variability means wider intervals), and confidence level (higher confidence requires wider intervals).
Common confidence levels and their uses:
- 90% Confidence Interval: Produces the narrowest interval but with less certainty. Used when a rough estimate is acceptable or when sample sizes are limited.
- 95% Confidence Interval: The most commonly used level in scientific research. Provides a good balance between precision and confidence.
- 99% Confidence Interval: Produces the widest interval but with the highest certainty. Used when the cost of being wrong is very high, such as in medical or safety-critical applications.
Formulas and Calculations
Confidence Interval for Mean (using Z-distribution):
CI = x-bar +/- z * (sigma / sqrt(n))
Where x-bar is the sample mean, z is the critical value from the standard normal distribution, sigma is the standard deviation, and n is the sample size. The term sigma/sqrt(n) is called the standard error of the mean.
Confidence Interval for Proportion:
CI = p-hat +/- z * sqrt( p-hat * (1 - p-hat) / n )
Where p-hat is the sample proportion (successes divided by sample size) and the square root term is the standard error of the proportion.
Margin of Error:
ME = z * Standard Error
The margin of error is half the width of the confidence interval. It tells you how far the interval extends on each side of the point estimate.
Critical Z-Values: 1.645 for 90% confidence, 1.96 for 95% confidence, and 2.576 for 99% confidence. These values correspond to the points on the standard normal distribution that cut off the specified percentage in the middle.