Confidence Interval Calculator

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Confidence Interval Calculator

Estimate the range where true population means and proportions likely fall.

Last updated: March 2026

Calculate Confidence Interval

95% Confidence Interval
[71.4215, 78.5785]
Standard Error
1.8257
Margin of Error
±3.5785
Z-score
1.960

What is a Confidence Interval?

A confidence interval is a range of values that estimates where a population parameter (like the true mean) likely falls. We can't measure entire populations, so we use sample data and apply statistical methods to create this range.

The confidence level (e.g., 95%) doesn't mean there's a 95% chance the true value is in this interval—it means that if we repeated the sampling process 100 times, approximately 95 of the constructed intervals would contain the true parameter. The interval is either right or wrong; the probability refers to the method, not the interval itself.

Confidence intervals are used everywhere: political polling ("The candidate will receive 52% ± 3% of votes"), medical research ("The new drug reduces symptoms by 20-30%"), and quality control ("Products average 100g ± 1g").

How to Calculate Confidence Intervals

For Means (Z-Interval)

SE = σ / √n
MOE = z* × SE
CI = [x̄ - MOE, x̄ + MOE]
Where z* is the critical value for your confidence level (1.96 for 95%)

For Proportions (p̂ ± MOE)

SE = √[p̂(1-p̂)/n]
MOE = z* × SE
CI = [p̂ - MOE, p̂ + MOE]
Where p̂ = x/n is the sample proportion

Z-Score Values (Standard Normal)

90% confidence: z = 1.645
95% confidence: z = 1.96
99% confidence: z = 2.576

Example: Survey of Test Scores

You survey 30 students and get a mean test score of 75 with a standard deviation of 10. What's the 95% confidence interval for the true population mean?

Given:
x̄ = 75, σ = 10, n = 30, confidence = 95%
Step 1:
SE = 10 / √30 = 10 / 5.477 = 1.826
Step 2:
z* = 1.96 (from 95% confidence level)
Step 3:
MOE = 1.96 × 1.826 = 3.58
Result:
CI = [75 - 3.58, 75 + 3.58] = [71.42, 78.58]

We're 95% confident the true mean test score is between 71.42 and 78.58.

Frequently Asked Questions

Why is the interval wider with higher confidence levels?

Higher confidence (99% vs 95%) uses a larger z-score, making the margin of error bigger. You're more certain the true value is in the range, but the range is wider.

How does sample size affect the interval?

Larger samples make the interval narrower (more precise). SE decreases with √n, so doubling sample size reduces margin of error by ~29%. More data = more precise estimates.

What does 95% confidence really mean?

It's about the method: if you repeated sampling and CI calculation 100 times, ~95 of those intervals would contain the true parameter. It's not about this one interval.

Can I choose my confidence level?

Yes, but there's always a tradeoff. 90% is narrower (less precise), 99% is wider. 95% is conventional. Use domain knowledge and precision requirements to choose.

What if my sample is too small?

This calculator uses z-scores (assumes large n, typically n≥30). For smaller samples, use t-scores instead, which are slightly larger to account for more uncertainty.

Does the CI mean the true value is probably inside?

Not quite. The true value either is or isn't in the interval (fixed, though unknown). The 95% refers to the method, not this interval's probability.

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