Confidence Interval Calculator
Margin of error and confidence interval for a sample mean, with the standard error and the sample size for half the margin.
Results
What this tool does
Use z = 1.96 for 95 % confidence, 1.645 for 90 % and 2.576 for 99 %. The normal z value assumes either a large sample or a known population standard deviation; below about thirty observations with an estimated deviation, the t distribution gives a wider and more honest interval.
Formula
IC = x̄ ± z × s ÷ √n
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mn | Mean (average) | — |
sd | Standard deviation (sample) | — |
nn | Sample size | — |
zz | Z value (confidence) | — |
HW | Margin of error | — |
LO | Lower bound | — |
HI | Upper bound | — |
SE | Standard error | — |
N2 | Sample size | — |
Worked example
- Mean (average)68.4
- Standard deviation (sample)5.2
- Sample size30
- Z value (confidence)1.96
- Margin of error1.860796
- Lower bound66.539204
- Upper bound70.260796
- Standard error0.949386
- Sample size120
Limitations
- Standard deviation and variance are calculated for a sample (dividing by n − 1). For a full population, divide by n instead.
- The result is an estimate based only on the values you type. Real situations often include factors this calculator does not know about.
- For work that must comply with a standard or be signed off, check the result against the applicable code and have it reviewed by a qualified engineer.
Frequently asked questions
What does 95 % confidence actually mean?
That if you repeated the whole sampling exercise many times, about 95 of every 100 intervals built this way would contain the true value. It does not mean there is a 95 percent chance the true value sits in this particular interval — the true value is fixed, it is the interval that moves from sample to sample. The last result shows what halving the margin costs: four times the sample, because precision improves only with the square root of n.