Confidence Interval Calculator
Calculate confidence intervals for means, proportions, and differences. Essential for statistical inference, research, and data analysis.
Calculator Input
Results
Interpretation
We are 95% confident that the true population mean lies between 67.6130 and 69.3870.
Step-by-Step Calculation
How to Use
Basic Steps
- Select the type of confidence interval you need
- Choose your desired confidence level (90%, 95%, 99%)
- Enter your sample statistics (mean, proportion, size, etc.)
- For two-sample tests, enter data for both samples
- Review the calculated interval and interpretation
Input Guidelines
- Sample size should be at least 30 for normal approximation
- Proportions should be between 0 and 1
- Use population SD when known, sample SD otherwise
- Higher confidence levels give wider intervals
Understanding Confidence Intervals
What is a Confidence Interval?
A confidence interval is a range of values that likely contains the true population parameter with a specified level of confidence.
Formula (for means):
SE = σ/√n or s/√n
Where SE is the standard error and the critical value depends on the confidence level.
Interpretation
A 95% confidence interval means that if we repeated the sampling process many times, 95% of the intervals would contain the true population parameter.
Key Concepts
Margin of Error
The maximum expected difference between the sample statistic and the true population parameter.
Critical Value
The Z-score or t-score that corresponds to the chosen confidence level and degrees of freedom.
Standard Error
The standard deviation of the sampling distribution, measuring the precision of the sample statistic.
Example Calculations
Example 1: Mean Height
Problem: Find a 95% CI for average height from a sample of 50 people.
Given: x̄ = 68.5 inches, s = 3.2 inches, n = 50
Critical value (t₄₉) ≈ 2.01
ME = 2.01 × 0.453 = 0.91
CI = 68.5 ± 0.91 = [67.59, 69.41]
Interpretation: We are 95% confident the true average height is between 67.59 and 69.41 inches.
Example 2: Proportion
Problem: Find a 95% CI for the proportion supporting a policy.
Given: p̂ = 0.62, n = 400
Critical value (Z) = 1.96
ME = 1.96 × 0.0243 = 0.048
CI = 0.62 ± 0.048 = [0.572, 0.668]
Interpretation: We are 95% confident that between 57.2% and 66.8% of the population supports the policy.
Frequently Asked Questions
What's the difference between Z and t distributions?
Use Z when the population standard deviation is known or sample size is large (n ≥ 30). Use t when the population standard deviation is unknown and sample size is small (n < 30).
How do I choose the confidence level?
95% is most common in research. Use 99% when you need higher confidence (but wider intervals). Use 90% when you can accept lower confidence for narrower intervals.
What if my confidence interval is very wide?
Wide intervals indicate high uncertainty. You can narrow them by increasing sample size, reducing confidence level, or using more precise measurement methods.