Confidence Interval Calculator
Calculate confidence intervals for a population mean or proportion — with the standard error, margin of error, step-by-step math, and a 90/95/99% comparison, all live as you type.
(95.1, 104.9)
CI Interval
±4.9
Margin of Error
95%
Confidence
CI for Population Mean
95% CI for Mean (μ)
(95.100, 104.900)
Width = 9.800 · MOE = ±4.900
- Margin of Error
- ±4.900
- Critical Value (z*)
- 1.960
95.100
Lower Bound
104.900
Upper Bound
2.500
Std Error
9.800
Width
Visual Confidence Interval
Lower100.000
Mean104.900
Upper
Compare Confidence Levels (90% / 95% / 99%)
A higher confidence level requires a wider interval to keep the same guarantee — precision and confidence trade off against each other for a fixed sample size.
Step-by-Step Solution
Standard Error (SE)
SE = s / √n = 15 / √36 = 2.5000
Critical value
z* for 95% = 1.960
Margin of Error
MOE = z* × SE = 1.960 × 2.5000 = 4.9000
CI bounds
Lower = 100 − 4.9000 = 95.1000 Upper = 100 + 4.9000 = 104.9000
Normal Distribution with CI Region
Standard deviations from the mean (z)
What Is a Confidence Interval?
A confidence interval is a range of values, computed from sample data, that is likely to contain the true population parameter — the mean or proportion you'd get if you could measure every single member of the population instead of just a sample. The confidence level attached to it (typically 90%, 95%, or 99%) describes how often the method used to build the interval succeeds in capturing that true value, not how likely any one interval is to be right.
This is the point most people get wrong: a “95% confidence interval” does not mean there is a 95% probability the true value sits inside the specific range you just calculated. Once your sample is collected and the interval is fixed, the true value is either inside it or it isn't — there's no probability left to assign. What 95% actually refers to is the long-run behavior of the method: if you drew a new random sample and rebuilt the interval a hundred times, roughly 95 of those hundred intervals would contain the true population value. The confidence lives in the procedure, evaluated across repeated sampling — not in the single interval sitting in front of you.
CI Formulas
Mean (σ known or n ≥ 30): CI = x̄ ± z* × (s/√n) [z-distribution]Mean (σ unknown, n < 30): CI = x̄ ± t* × (s/√n) [t-distribution, df = n − 1]Proportion: CI = p̂ ± z* × √(p̂(1 − p̂)/n)Common critical values (z*): 90% = 1.645 | 95% = 1.96 | 99% = 2.576Note on proportions: because a proportion can never fall outside 0 and 1, this calculator clamps the interval to that range when a very wide margin of error (typically from a small sample) would otherwise push the raw bounds past it.
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Standard Error
How much a sample statistic is expected to vary from sample to sample — it shrinks as n grows, which is why bigger samples give narrower intervals.
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Margin of Error
The critical value times the standard error — this is the ± amount added and subtracted from your sample statistic to build the interval.
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Critical Value (z*)
How many standard errors wide the interval needs to be to hit the target confidence level, read from the standard normal distribution.
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Repeated Sampling
The 95% figure describes the method across many hypothetical samples, not the probability the true value sits in this one interval.
— FAQ