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Confidence Interval Upper Limit

U=xˉ+zσnU = \bar{x} + z \frac{\sigma}{\sqrt{n}}

Units aren’t used in this calculation: every value is a plain number.

sig figs

Worked example: mean 50, 95% CI, sigma 10, n 100 → U = 51.96. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

Where Confidence Interval Upper Limit fits

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See also

Confidence Interval Lower Limit: Both use sample mean and critical z-value.Margin of Error for a Mean: Both use critical z-value and standard deviation.Sample Size for a Mean: Both use critical z-value and standard deviation.One-Sample T-Test Statistic: Both use sample mean and standard deviation.One-Sample Z-Test Statistic: Both use sample mean and standard deviation.Two-Sample Z-Test Statistic: Both use sample mean and standard deviation.
6 more relatedMargin of Error for a Proportion: Both use critical z-value and sample size.Sample Size for a Proportion: Both use critical z-value and sample size.Weighted Mean of Two Groups: Both use sample mean and sample size.Cohen's d (Effect Size): Both use sample mean.Regression Line Intercept: Both use sample mean.Arithmetic Mean of Two Numbers: Both use sample mean.
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Confidence Interval Upper Limit explained

x̄Uzσ/√n

The upper confidence limit is the sample mean pushed up by z standard errors, and it answers the question regulators and engineers actually ask: how high could the true mean plausibly be? A batch of 100 samples averaging 50.0 with σ = 10 has a 95% upper limit of 50.0 + 1.96 × 1 = 51.96. Environmental permits, contaminant screening and food-safety limits are almost always written against an upper confidence limit rather than a point estimate, precisely because being wrong in that direction is the expensive kind of wrong.

Two things to watch. Widening the confidence level always pushes the upper limit further out: going from 95% (z = 1.96) to 99% (z = 2.576) buys certainty by admitting you know less about where μ sits. And the limit is a statement about the mean, not about individual observations; roughly half the individual values will exceed the upper confidence limit for the mean, which is a genuinely surprising fact the first time you meet it. Solving for z reverses the calculation: if a reported interval on a mean of 80 tops out at 84 with σ = 15 and n = 36, the critical value used was z = (84 − 80) × 6/15 = 1.6, about 89% confidence.

Confidence Interval Upper Limit formula

U=xˉ+zσnU = \bar{x} + z \frac{\sigma}{\sqrt{n}}
Where
  • UU= Upper confidence limit
  • xˉ\bar{x}= Sample mean
  • zz= Critical z-value
  • σ\sigma= Standard deviation
  • nn= Sample size
In this toolkit

Confidence intervals and sample size

8 formulasStandard error, margin of error and the sample size a target precision demands — for both means and proportions.

Revised 31 October 2026
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formula.expert. (2026, October 31). Confidence Interval Upper Limit Formula & Calculator. https://formula.expert/solvers/confidence-interval-upper-limit

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