Confidence Interval Calculator Mean, z/t
Real-time CI for a population mean using z (when σ known) or t (when σ unknown & using sample s).
📊 Sample mean (x̄)
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Enter the observed average of your sample.
🔧 Dispersion type
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This matches common practice: z with known σ; t with sample s.
📐 Standard deviation (σ or s)
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Non-negative.
👥 Sample size (n)
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Integer ≥ 1.
🎯 Confidence level (%)
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Two-sided; between 50 and 99.999.
⚡ Presets
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Optional helper; you can still type a custom % above.
Results
Standard error (SE = σ/√n or s/√n)
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Critical value (z* or t*)
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Margin of error (ME = critical × SE)
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Lower bound
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Upper bound
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Show step-by-step derivation
References & Notes
- Confidence interval formulas for a mean with known population σ (z) and unknown σ using sample s (t) are standard results in intro statistics texts.
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Critical values:
- z*: two-tailed quantile of N(0,1) at CL/2 in each tail.
- t*: two-tailed quantile of tdf=n−1 at CL/2 in each tail.