Confidence interval calculator: build a confidence interval for a mean from sample size, mean, and standard deviation. Free, private, runs in your browser.
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Confidence interval calculator: build a confidence interval for a mean from sample size, mean, and standard deviation. Free, private, runs in your browser.
95% interval [11.224, 13.576] with a margin of error of 1.176 mm.
Transparent methodology
How this calculator works
Reviewed August 2026 · BoringToolsKit Editorial Team
Formula
CI = mean +/- z* x sd / sqrt(n); z* = 1.96 for 95% confidence.
Worked example
Mean 100, standard deviation 15, n = 36: standard error 2.5, margin of error +/-4.9, a 95% interval of [95.1, 104.9].
BoringToolsKit. “Confidence Interval Calculator.” boringtoolskit.com/confidence-interval-calculator/ (reviewed August 2026). Free to reference in articles, syllabi, and answer posts with a link.
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Worked example: from inputs to interval in three steps
Say 36 bearings were sampled, with a sample mean of 12.4 mm and a sample standard deviation of 3.6 mm, at 95% confidence. Step 1, standard error: se = 3.6 / sqrt(36) = 3.6 / 6 = 0.6 mm. Step 2, margin of error: 1.96 x 0.6 = 1.176 mm. Step 3, the interval: 12.4 - 1.176 = 11.224 and 12.4 + 1.176 = 13.576, so the answer shown is [11.224, 13.576] with a margin of error of plus or minus 1.176 mm.
The same data at 90% confidence gives 1.645 x 0.6 = 0.987, so [11.413, 13.387]. At 99% it gives 2.576 x 0.6 = 1.546, so [10.854, 13.946]. Notice the trade: raising confidence from 95% to 99% widens the interval by 0.370 mm on each side. Doubling the sample size instead (n = 72, same sd) cuts the standard error to 0.424 and the 95% margin to 0.832. Precision bought with confidence costs width; precision bought with sample size costs money and time, not width.
Where the z* shortcut bends the truth
The z* multipliers (1.645, 1.96, 2.576) are exact for a normal population only when the standard deviation is known. When you estimate sd from the sample itself, the honest multiplier is t with n-1 degrees of freedom. For n = 6, the 95% t value is 2.571, not 1.96: with sd = 3.6 the true margin is 2.571 x (3.6 / sqrt(6)) = 3.779, versus 2.883 from the z shortcut, roughly 31% too narrow. By n = 31 the t value is 2.042, about 4% above 1.96, and by n = 100 it is 1.984, close enough to ignore.
A second limit: the interval covers the population mean in 95% of repeated samples. It is not a 95% probability that any one computed interval contains the mean, and it says nothing about individual values. For that you need a prediction or tolerance interval, which is roughly z x sd x sqrt(1 + 1/n) wide, about 3.6 x 2.24 = 8.07 mm for the bearing example, far wider than the 2.352 mm mean interval. Also remember the validity gate at n < 2: with one observation there is no way to estimate spread, so no interval exists at all.
