Confidence Interval Calculator
Intervals for means and proportions, with margin of error, so you can see how precise a sample estimate is.
Confidence Interval Calculator
Calculate confidence intervals for population means and proportions with detailed statistical analysis
Example Scenarios
What a confidence interval is
A confidence interval is a range that, under repeated sampling, would cover the true population parameter at a stated rate (e.g. 95%). If one sample gives [45.2, 52.8] at 95%, that does not mean there is a 95% probability the mean sits in that specific interval—it means the procedure catches the truth about 95% of the time across many samples.
Mean (μ)
CI = x̄ ± ME
ME = t(α/2) × (s/√n)
Use t when σ is unknown (usual case).
Proportion (p)
CI = p̂ ± ME
ME = z(α/2) × √(p̂(1−p̂)/n)
Need np̂ ≥ 5 and n(1−p̂) ≥ 5 for the normal approx.
Confidence levels
Higher confidence → wider intervals. 95% (z ≈ ±1.96) is the default tradeoff; 90% is tighter (z ≈ ±1.645); 99% is wider (z ≈ ±2.576).
Exploratory, when you favor narrow width
Standard reporting choice
High-stakes / conservative work
Common formulas
Mean, σ known
x̄ ± z(α/2) × (σ/√n)
Mean, σ unknown
x̄ ± t(α/2, n−1) × (s/√n)
Proportion
p̂ ± z(α/2) × √(p̂(1−p̂)/n)
What widens or narrows the interval
Wider
- Higher confidence level
- Larger standard deviation
- Smaller n
Narrower
- Lower confidence level
- Less variability
- Larger n (ME scales like 1/√n—quadruple n to halve ME)
Worked examples
Mean (σ unknown): n=25, x̄=78.4, s=12.6, 95%
df=24, t≈2.064 → ME = 2.064×(12.6/5) ≈ 5.20 → CI [73.2, 83.6]
Proportion: 240/400 = 0.60, 95%
ME = 1.96×√(0.24/400) ≈ 0.048 → CI [0.552, 0.648]
Pitfalls
Misreading 95%
It is about the method under repetition, not a probability for one fixed interval.
Wrong critical value
Use t when σ is estimated from the sample; z for known σ or large-sample proportions with conditions met.
Check assumptions
Independence, approximate normality (or large n), representative sample.
Report fully
Level, n, and a plain-language interpretation—and ask if the width is useful.
Frequently Asked Questions
What does "95% confident" mean?
Across many samples, about 95% of the intervals you build this way would contain the true parameter. One computed interval either does or does not.
t vs z?
t when σ is unknown and estimated by s. z when σ is known, or for proportions with adequate counts.
How large should n be?
Roughly n = (z σ / E)² for a desired margin E (use a pilot estimate of σ). For proportions: n = z² p(1−p) / E².
Interval too wide?
Grow n (most effective), accept a lower confidence level, or reduce noise in the measurement.
Interval outside [0, 1] for a proportion?
Clip to the [0, 1] range for reporting, or use a better method (Wilson, etc.) when p̂ is near 0 or 1 or n is small.
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