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Confidence Interval Calculator

Math Calculators

Mean CI at 90/95/99% with margin of error

Quick Answer

Confidence Interval Calculator: Mean CI at 90/95/99% with margin of error. Point estimate: sample mean x̄.

95% confidence interval

[95.100, 104.900]

100 ± 4.900（margin of error）

Std error SE2.5000
Z score1.96
Margin ±4.9000
n for half margin144

Steps

  1. 1SE = σ/√n = 15/√36 = 2.5000
  2. 295% → Z = 1.96
  3. 3Margin = Z × SE = 1.96 × 2.5000 = 4.9000
  4. 4CI = 100 ± 4.9000 = [95.100, 104.900]
  5. 5To halve the margin, need n = (Z·σ/E)² = 144 (4× not 2×!)

Interval position around the mean

outside CI
95% CI
outside CI

Mean 100 at center; width = ±4.900

Sample size vs margin (same σ, level)

nMargin ±vs current
255.8801.2× narrower
504.1581.2× wider
1002.9401.7× wider
4001.4703.3× wider
10000.9305.3× wider
25000.5888.3× wider
Margin shrinks with √n, not n: quadrupling the sample only halves the margin. That is why national polls stop at n≈1000 — beyond that, precision gains are marginal while costs grow linearly.

Z values reference

80%1.282
90%1.645
95% ←1.96
98%2.326
99%2.576
99.9%3.291

💡 Correct interpretation: "95% of intervals built this way contain the true mean" — not "95% probability the mean is here". Three counterintuitive facts: halving the margin needs 4× the sample; past n≈120 returns diminish fast; for proportions the worst-case variance is at p=0.5, so polls with n=1000 carry ±3.1% max margin at 95%.

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How to Use

  1. Enter values or expressions: type directly into the Confidence Interval Calculator — keyboard friendly.
  2. Instant results: everything computes live as you type, no button needed.
  3. Follow the steps: the result panel shows full working and formula substitution.
  4. Copy or recompute: one-click copy; change any input to recalculate instantly.

Key Points

  • Confidence Interval Calculator is a free online math tool — open and use, no sign-up.
  • All calculations run locally in your browser; your data never leaves the device.
  • Key capability: Mean CI at 90/95/99% with margin of error.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Confidence interval for a population mean

Point estimate: sample mean x̄
Standard error: SE = σ / √n (use s if σ unknown)
CI = x̄ ± z* × SE
Common z*: 90% → 1.645; 95% → 1.96; 99% → 2.576
Larger n narrows the interval (1/√n scaling)

Example: n=100, x̄=500g, σ=15g, 95% CI = 500 ± 1.96×1.5 = [497.06, 502.94] g.

FAQ

How do I use the Confidence Interval Calculator?
Open the page and fill in the input fields — results update automatically. Mean CI at 90/95/99% with margin of error.
Are the results accurate?
Yes. All calculations use standard mathematical formulas and are verified for accuracy.
Is it free? Do I need to sign up?
Completely free. No registration, login or download required — just open and use.
Does it work on mobile?
Yes. The responsive design works on phones, tablets and desktops.
Is my data saved?
No. All calculations run locally in your browser. Nothing is uploaded or stored.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated:

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

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Cite this page: YiJianSuan. Confidence Interval Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/confidence-interval

Results provided by this tool are for reference only and do not constitute professional advice.