Confidence Interval Calculator - Calculate CI for Mean & Proportion
Calculate confidence intervals for population mean and proportion with margin of error, upper and lower bounds for statistical estimation
Enter Sample Statistics
Confidence Interval
Interpretation
What is a Confidence Interval Calculator?
A Confidence Interval Calculator estimates a range of values likely to contain a population parameter with a specified level of confidence, providing both bounds and margin of error.
This calculator is used for:
- Population Estimation - Estimate population parameters from samples
- Hypothesis Testing - Statistical inference and decision making
- Research Analysis - Report uncertainty in findings
- Quality Control - Process capability assessment
To understand the theoretical foundation of confidence intervals, explore our Central Limit Theorem Calculator to see how sampling distributions enable interval estimation.
To work with the empirical rule for normal distributions, check out our Empirical Rule Calculator to understand how data falls within standard deviations.
To calculate probabilities for discrete outcomes, visit our Binomial Distribution Calculator to determine confidence intervals for proportions in binary experiments.
To test hypotheses about categorical data, use our Chi-Square Calculator to determine if observed frequencies differ from expected values.
How Confidence Intervals Work
Confidence interval formulas:
Where Z is critical value from standard normal or t-distribution.
Key Concepts Explained
Confidence Level
The probability that the interval contains the true parameter. Higher confidence levels produce wider intervals.
Margin of Error
Half the interval width, representing maximum expected deviation from the true value.
Standard Error
Measures sampling variability. Smaller SE leads to narrower, more precise intervals.
How to Use This Calculator
Select Interval Type
Choose Mean or Proportion interval
Enter Sample Statistics
Input mean/proportion, standard deviation, and sample size
Choose Confidence Level
Select 90%, 95%, or 99% confidence level
Benefits of This Calculator
- Multiple Types - Mean and proportion intervals
- Instant Results - Immediate calculation of bounds
- Clear Interpretation - Plain English explanation
- Statistical Rigor - Accurate critical values
- Educational Tool - Learn about confidence intervals
- Research Ready - Professional statistical analysis
Factors Affecting Results
- Sample Size - Larger samples produce narrower intervals
- Confidence Level - Higher confidence widens interval
- Variability - Higher standard deviation widens interval
- Distribution - Use t for small samples or unknown σ
- Sampling Method - Random sampling is essential
- Population Size - Finite correction for small populations
Frequently Asked Questions
What is a confidence interval?
A confidence interval is a range of values that is likely to contain an unknown population parameter with a specified level of confidence. For example, a 95% confidence interval means we're 95% confident the true parameter lies within that range.
How do you calculate a confidence interval?
For means: CI = X̄ ± (Z × SE), where X̄ is sample mean, Z is the critical value, and SE is standard error (σ/√n). For proportions: CI = p̂ ± (Z × √(p̂(1-p̂)/n)).
What does 95% confidence interval mean?
A 95% confidence interval means that if we repeated the sampling process many times, approximately 95% of the intervals would contain the true population parameter. It reflects our uncertainty about the estimate.
What is the margin of error in confidence intervals?
The margin of error is half the width of the confidence interval, calculated as Z × SE. It represents the maximum expected difference between the sample estimate and the true population parameter.
When should you use a t-distribution vs Z-distribution?
Use t-distribution when sample size is small (n < 30) or population standard deviation is unknown. Use Z-distribution when n ≥ 30 or population standard deviation is known.
How does sample size affect confidence intervals?
Larger sample sizes produce narrower confidence intervals, meaning more precise estimates. As n increases, the standard error decreases (SE = σ/√n), reducing the margin of error.