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Normal Distribution Calculator

Math Calculators

Z-score, percentile and probability between values

Quick Answer

Normal Distribution Calculator: Z-score, percentile and probability between values. z = (x − μ) / σ — how many standard deviations from the mean.

P(X ≤ 1.96)

97.5002%

Z = (1.96 − 0) / 1 = 1.9600

Z 分数1.9600
Below97.5002%
Above2.4998%

Steps

  1. 1Standardize: Z = (x − μ) / σ
  2. 2Z = (1.96 − 0) / 1 = 1.9600
  3. 3Φ(1.9600) = 97.5002%

Empirical rule (68-95-99.7)

±1σ 68%
±2σ 95%
±3σ 99.7%

|Z| = 1.96

σ intervals

μ ± 1σ68.27%
μ ± 2σ95.45%
μ ± 3σ99.73%
μ ± 4σ99.994%

Common Z values

z = 0Φ(z) = 50.00%
z = 0.5Φ(z) = 69.15%
z = 1Φ(z) = 84.13%
z = 1.28Φ(z) = 90.00%
z = 1.645Φ(z) = 95.00%
z = 1.96Φ(z) = 97.50%
z = 2Φ(z) = 97.72%
z = 2.326Φ(z) = 99.00%
z = 2.576Φ(z) = 99.50%

💡 CDF computed via Abramowitz–Stegun erf approximation (error ≤ 1.5e-7); quantiles via Acklam rational approximation. Plenty for statistics homework and quality control.

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How to Use

  1. Enter values or expressions: type directly into the Normal Distribution 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

  • Normal Distribution 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: Z-score, percentile and probability between values.
  • Results come with formulas and reference tables so you can verify every number.

Formula & How It Works

Normal distribution & z-score

z = (x − μ) / σ — how many standard deviations from the mean
P(X<x) = Φ(z); P(X>x) = 1 − Φ(z); P(a<X<b) = Φ(z_b) − Φ(z_a)
68-95-99.7 rule: μ±1σ covers 68.3%, ±2σ 95.4%, ±3σ 99.7%

Example: μ=75, σ=10, score 92 → z = 1.7 → Φ(1.7) ≈ 0.9554, beating 95.5% of test-takers.

FAQ

How do I use the Normal Distribution Calculator?
Open the page and fill in the input fields — results update automatically. Z-score, percentile and probability between values.
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:

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