Confidence Interval Builder
Calculate confidence intervals for means and proportions with visualization. Enter values for instant results with step-by-step formulas.
Formula
CI = xฬ ยฑ z ร (ฯ/โn)
The confidence interval equals the sample mean plus/minus the z-score times the standard error. Standard error is population standard deviation divided by square root of sample size.
Worked Examples
Example 1: Survey Satisfaction Score
Problem:A survey of 200 customers shows mean satisfaction of 7.5/10 with SD=1.8. Calculate 95% CI.
Solution:Given: xฬ=7.5, n=200, ฯ=1.8, z=1.96\n\nSE = 1.8/โ200 = 0.127\nMoE = 1.96 ร 0.127 = 0.249\n\n95% CI = 7.5 ยฑ 0.249\n= (7.251, 7.749)\n\nInterpretation: We're 95% confident true mean satisfaction is between 7.25 and 7.75.
Result:95% CI: (7.25, 7.75) | MoE = 0.25 points
Example 2: Election Poll
Problem:Poll of 1,000 voters shows 52% support candidate A. Calculate 95% CI for proportion.
Solution:Given: pฬ=0.52, n=1000, z=1.96\n\nSE = โ(0.52ร0.48/1000) = 0.0158\nMoE = 1.96 ร 0.0158 = 0.031\n\n95% CI = 0.52 ยฑ 0.031\n= (0.489, 0.551) or 48.9% to 55.1%\n\nSince CI includes 50%, result is not statistically conclusive.
Result:95% CI: (48.9%, 55.1%) | Too close to call
Example 3: Manufacturing Quality
Problem:Sample of 50 parts has mean length 10.02mm, SD=0.05mm. Find 99% CI.
Solution:Given: xฬ=10.02, n=50, ฯ=0.05, z=2.576\n\nSE = 0.05/โ50 = 0.00707\nMoE = 2.576 ร 0.00707 = 0.0182\n\n99% CI = 10.02 ยฑ 0.0182\n= (10.0018, 10.0382)\n\nNarrower than typical due to low variance.
Result:99% CI: (10.002, 10.038) | Precision ยฑ0.018mm
Frequently Asked Questions
What is a confidence interval?
A confidence interval is a range of values that likely contains the true population parameter. A 95% CI means if we repeated the sampling process many times, 95% of the calculated intervals would contain the true value. It quantifies uncertainty in our estimate.
What does 95% confidence level mean?
95% confidence means that if you were to take 100 different samples and compute a CI for each, approximately 95 of those intervals would contain the true population parameter. It does NOT mean there's a 95% probability the true value is in this specific interval.
How do I interpret overlapping confidence intervals?
Overlapping CIs suggest the difference may not be statistically significant, but this is not a definitive test. Two groups with overlapping 95% CIs can still have p < 0.05 difference. Use formal hypothesis tests for comparison.
What affects confidence interval width?
CI width is determined by: 1) Confidence level (higher = wider), 2) Sample size (larger = narrower), 3) Population variability (more variable = wider), 4) Distribution (t wider than z for small n).