Confidence Interval Calculator
Calculate confidence intervals for population means and sample proportions with custom confidence levels ($90\%, 95\%, 99\%$), critical values ($z^*, t^*$), standard errors, and interactive shaded normal bell curves.
Sample Parameters
Step-by-Step Margin of Error & Boundary Proof
How to Calculate a Confidence Interval
To find a confidence interval: 1. Calculate Standard Error: SE = s / √n (or √[p(1-p)/n]). 2. Look up critical value for chosen confidence level (e.g. z* = 1.96 for 95%). 3. Compute Margin of Error: ME = Critical Value × SE. 4. Confidence Interval = [Point Estimate − ME, Point Estimate + ME].
Confidence Interval Definition & Frequentist Interpretation
In inferential statistics, a Confidence Interval (CI) provides an estimated range likely to contain an unknown population parameter with a quantifiable degree of certainty:
The single best empirical guess of the true population parameter from your sample.
The maximum expected sampling discrepancy bounded by the chosen confidence level.
Z-Interval vs. Student’s t-Interval Comparison
| Interval Type | When Applicable | Critical Value Metric |
|---|---|---|
| Z-Interval | Population σ is known, or large sample size n ≥ 30 | z* (Standard Normal distribution) |
| Student's t-Interval | Population σ unknown (use sample s), small n < 30 | t* (Student-t with df = n − 1) |
| Proportion Interval | Binomial survey data (np ≥ 10 and n(1-p) ≥ 10) | z* with SE = √[p̂(1−p̂)/n] |
Real-World Applications of Confidence Intervals
Political Election Polling
Polling agencies report candidate support as 52% ± 3.1% (95% CI: [48.9%, 55.1%]), showing whether the lead is statistically significant.
Pharmaceutical Clinical Trials
FDA regulators require 95% confidence intervals for drug efficacy to prove cholesterol reductions outperform placebo baselines.
A/B Testing in Software
Tech companies calculate conversion rate confidence intervals to ensure UI redesign revenue lifts are not random variance.
Step-by-Step Worked Numerical Solutions
Problem: A random sample of 64 students has mean IQ x̄ = 104 with population SD σ = 15. Find the 95% confidence interval.
Common Pitfalls & Mistakes
Misinterpreting 95% Probability
The 95% applies to the sampling procedure across repeated studies, NOT the probability of the fixed population mean.
Using Z Instead of t on Small Samples
When σ is unknown and n < 30, using z* instead of t* underestimates the margin of error and produces overly narrow intervals.
Dividing by n Instead of √n
Standard error requires the square root of sample size √n in the denominator, not n.
Lead Developer & Founder of Basic Math Tools. Specializes in browser-native computational algorithms and applied mathematics.
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