Confidence Interval Calculator
Given a sample mean, standard deviation and sample size, compute the confidence interval of the population mean: X̄ ± z·σ/√n. Choose a 90%, 95% or 99% confidence level and see the margin of error and both bounds with full steps.
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95% Confidence Interval
19.713 — 21.487
X̄ = 20.6 ± 0.887
- Sample size (n)
- 50
- Sample mean (X̄)
- 20.6
- Std dev (σ or s)
- 3.2
- z for 95%
- 1.96
- Standard error (σ/√n)
- 0.45255
- Margin of error
- ± 0.88698
- Lower bound
- 19.71302
- Upper bound
- 21.48698
- Margin as % of mean
- 4.31%
- Interval width
- 1.77396
Step-by-step solution
- 1Find the critical z valuez = normInv(1 − (1 − 95%)/2) = normInv(0.975) = 1.96
- 2Standard errorSE = σ/√n = 3.2 / √50 = 3.2 / 7.0711 = 0.45255
- 3Margin of errorME = z × SE = 1.96 × 0.45255 = 0.88698
- 4Build the intervalX̄ ± ME = 20.6 ± 0.88698 → (19.71302, 21.48698)
This z-interval assumes the sample mean is approximately normally distributed (fine for n ≥ 30 by the central limit theorem) and σ is known or well-estimated. For small samples with unknown σ, a t-interval is more appropriate.
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Formula
CI = X̄ ± z × σ/√n, where z = normInv(1 − (1 − CL)/2)
- • X̄ = sample mean
- • σ or s = standard deviation
- • n = sample size
- • z = critical value for the chosen confidence level
Pro tips
- • Quadrupling the sample size halves the margin of error.
- • Higher confidence → wider interval; there is no free lunch.
- • If you only have raw data, compute its mean and standard deviation first (see Statistics Calculator).
FAQ
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