Confidence Interval Calculator

Mean & Proportion Two-Sample t-Distribution Free · No Signup

Free online confidence interval calculator for means and proportions. Compute one-sample and two-sample CIs with t-distribution or z-score, margin of error, step-by-step KaTeX formulas, interactive Plotly chart, and Python scipy export.

Confidence Interval
Average of your sample data
Uses t-distribution with n−1 degrees of freedom

Result

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Enter parameters and click Calculate

Compute confidence intervals for means, proportions, or differences.

CI Visualization

Python Compiler

What Is a Confidence Interval?

A confidence interval (CI) is a range of values that likely contains the true population parameter. It quantifies the uncertainty in a sample estimate and is fundamental to statistical inference.

Lower x̄ Upper 95% Confidence Interval
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Point Estimate

The sample statistic (mean or proportion) is our best single guess for the population parameter.

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Margin of Error

How far the interval extends from the point estimate. Depends on confidence level, variability, and sample size.

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Interval Width

Narrower = more precise. Increase sample size or decrease confidence level for a tighter interval.

Confidence Interval Formulas

Mean (one-sample):  CI = x̄ ± tα/2, n−1 × s / √n
Proportion:  CI = p̂ ± zα/2 × √(p̂(1−p̂) / n)
Difference in means:  CI = (x̄1 − x̄2) ± t × √(s1²/n1 + s2²/n2)
Difference in proportions:  CI = (p̂1 − p̂2) ± z × √(p̂1(1−p̂1)/n1 + p̂2(1−p̂2)/n2)
Worked Example: A sample of n=25 students has mean score x̄=78 and s=8. Find the 95% CI for the population mean.
SE = 8 / √25 = 1.6
df = 24, t0.025, 24 = 2.064
MoE = 2.064 × 1.6 = 3.30
95% CI = [78 − 3.30, 78 + 3.30] = [74.70, 81.30]

Common Confidence Levels

Levelz* Critical ValueUse Case
90%1.645Quick estimates, exploratory analysis
95%1.960Standard for most research and publications
99%2.576High-stakes decisions, critical applications

Trade-off: Higher confidence level → wider interval (more certainty, less precision). Larger sample size → narrower interval (more precision without sacrificing certainty).

Interpreting Confidence Intervals

✅ Correct Interpretation

“We are 95% confident that the true population mean lies between 45 and 55.” This refers to the procedure, not this specific interval.

❌ Common Mistake

“There is a 95% probability the true mean is in this interval.” The true mean is fixed — it either is or isn’t in the interval.

🔍 Two-Sample CIs

If a CI for a difference contains 0, the difference is not statistically significant. If it excludes 0, the groups differ significantly.

⚠️ Overlapping CIs

Two overlapping CIs do not necessarily mean no significant difference. Always use a proper two-sample test or CI.

Frequently Asked Questions

A 95% CI means that if you repeated the sampling process many times, 95% of the resulting intervals would contain the true population parameter. It does not mean there is a 95% probability this specific interval contains the true value.
Use the t-distribution for means when the population standard deviation is unknown (almost always). The z-score is used for proportions and when n is very large. For small samples, the t-distribution has heavier tails, giving wider intervals.
Larger samples reduce the standard error, which narrows the CI. The standard error decreases proportionally to √n, so quadrupling the sample size halves the interval width.
A one-sample CI estimates a single population parameter (mean or proportion). A two-sample CI estimates the difference between two population parameters — useful for comparing groups like treatment vs. control.
The Welch-Satterthwaite formula calculates effective degrees of freedom when comparing two means with unequal variances. It gives a more accurate t-critical value than the conservative min(n1−1, n2−1).
If a CI for the difference between two means or proportions includes zero, the difference is not statistically significant at that confidence level. The data is consistent with no real difference between the groups.

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