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Confidence Interval Lower Limit

L=xˉ−zσnL = \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 → L = 48.04. 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.

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See also

Confidence Interval Upper 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 Lower Limit explained

x̄Lzσ/√n

A confidence interval is the sample mean flanked by a margin of error, and this is its lower edge: back off from x̄ by z standard errors. A sample of 100 parts averaging 50.0 mm with σ = 10 mm gives a 95% lower limit of 50.0 − 1.96 × (10/10) = 48.04 mm. Jerzy Neyman set the logic out in 1937, and he was careful about what it claims: the procedure captures the true mean 95% of the time over many repetitions. Any one interval either contains μ or it does not.

That is the trap almost everyone falls into: reading "95% confidence" as "there is a 95% chance μ lies in here". The randomness lives in the interval, not in μ. In practice the lower limit is often the number that matters commercially: if a supplier guarantees a mean tensile strength above 48 MPa and your 95% lower limit lands at 48.04, you have just barely cleared the bar. Run the formula backwards to size a study: to keep the lower limit within 4 units of a mean of 100 when σ = 20 at z = 2, you need n = (2 × 20/4)² = 100.

Confidence Interval Lower Limit formula

L=xˉ−zσnL = \bar{x} - z \frac{\sigma}{\sqrt{n}}
Where
  • LL= Lower 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 Lower Limit Formula & Calculator. https://formula.expert/solvers/confidence-interval-lower-limit

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