Updated June 17, 2026
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A 95% confidence interval for a mean is: CI = x-bar plus or minus (1.96 * sigma / sqrt(n)), where x-bar is the sample mean, sigma is the standard deviation, and n is the sample size. The 1.96 comes from the normal distribution's critical value for 95% coverage.
The formula
CI = x_bar +/- z * (sigma / sqrt(n))
- x_bar: sample mean
- z: critical value (1.96 for 95%, 2.576 for 99%)
- sigma: population SD (use sample SD if population SD is unknown)
- n: sample size
- sigma / sqrt(n): the standard error of the mean
When you do not know the population SD and your sample is small, use the t-distribution instead. The standard deviation calculator gives you the SD you need.
Worked example: 95% confidence interval for a mean
You measure reaction times for 50 participants. Sample mean = 280 ms, sample SD = 40 ms.
| Step | Calculation | Value |
|---|---|---|
| Standard error | 40 / sqrt(50) | 5.66 ms |
| Margin of error (95%) | 1.96 * 5.66 | 11.09 ms |
| Lower bound | 280 - 11.09 | 268.9 ms |
| Upper bound | 280 + 11.09 | 291.1 ms |
The 95% confidence interval is 268.9 ms to 291.1 ms. You are 95% confident the true population mean reaction time falls in that range.
Where does 1.96 come from?
In the standard normal distribution (mean = 0, SD = 1), 95% of values fall between -1.96 and +1.96. This is a property of the bell curve: going 1.96 SDs in each direction from the center captures exactly 95% of the area under the curve. For 99% coverage, the comparable value is 2.576. These numbers come from z-score tables (also called normal distribution tables).
What does "95% confident" actually mean?
It does not mean there is a 95% chance the true value is in this specific interval. Rather, if you repeated the study 100 times with different samples, about 95 of the resulting intervals would contain the true population value. Any single interval either captures the true value or it does not; the 95% refers to the long-run reliability of the method, not this one interval. This is a subtle but important distinction in interpreting statistics.
How to find a 95% CI in Excel
Compute the margin of error with =CONFIDENCE.NORM(0.05, standard_dev, n), then subtract and add it to the mean. For small samples or unknown population SD, use =CONFIDENCE.T(0.05, standard_dev, n) instead. The z-score article covers how the critical value connects to the distribution.
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