Math & Engineering
Confidence Interval Calculator
Calculate confidence intervals to estimate population parameters from sample data
Enter values to calculate the confidence interval
Related to Confidence Interval Calculator
The confidence interval calculator determines a range of values that is likely to contain the true population parameter based on sample data. It uses the sample mean, standard error, and desired confidence level to calculate the interval. The formula used is:
Confidence Interval Formula
CI = Sample Mean ± (z-score × Standard Error)
where z-score depends on the confidence level: - 90% confidence: z = 1.645 - 95% confidence: z = 1.96 - 99% confidence: z = 2.576
The standard error represents the standard deviation of the sampling distribution. A smaller standard error indicates more precise estimates, while a larger standard error suggests more uncertainty in the estimation.
The confidence interval provides a range of values that likely contains the true population parameter. The width of this interval is determined by the margin of error, which is influenced by both the confidence level and the standard error.
Understanding the Output
- Lower Bound: The minimum value of the interval - Upper Bound: The maximum value of the interval - Margin of Error: The ± value that creates the interval around the sample mean
A wider interval suggests less precision but higher confidence, while a narrower interval indicates more precision but potentially lower confidence. The chosen confidence level (90%, 95%, or 99%) represents how often the interval would contain the true population parameter if the sampling process were repeated many times.
1. What is a confidence interval?
A confidence interval is a range of values that likely contains an unknown population parameter. It provides both an estimate of the parameter and the uncertainty associated with that estimate.
2. How do I choose the right confidence level?
The choice of confidence level depends on your needs. 95% is most common in research, 90% provides narrower intervals but less confidence, and 99% offers the highest confidence but wider intervals. Consider the balance between precision and confidence for your specific application.
3. What affects the width of a confidence interval?
The width of a confidence interval is affected by three main factors: the confidence level (higher levels create wider intervals), the standard error (larger errors create wider intervals), and the sample size (larger samples typically lead to narrower intervals).
4. Can I use this calculator for any type of data?
This calculator assumes your data follows a normal distribution and that you have already calculated the sample mean and standard error. It's most appropriate for continuous data with a roughly symmetric distribution.
5. What is the scientific source for this calculator?
This calculator implements the standard normal distribution-based confidence interval methodology as described in fundamental statistical theory. The z-scores used are derived from the standard normal distribution table and are widely accepted in statistical practice. The methodology is based on the Central Limit Theorem and is documented in standard statistical textbooks such as "Statistical Inference" by Casella and Berger, and "Introduction to Mathematical Statistics" by Hogg, McKean, and Craig. The implementation follows the guidelines established by the American Statistical Association (ASA) for statistical intervals.