Normal Distribution Calculator

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.

Mean μ

Std dev σ

P(X ≤ 1.96)

97.5002%

Z = (1.96 − 0) / 1 = 1.9600

Z 分数1.9600

Below97.5002%

Above2.4998%

Steps

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

Empirical rule (68-95-99.7)

±1σ 68%

±2σ 95%

±3σ 99.7%

|Z| = 1.96

σ intervals

μ ± 1σ	68.27%
μ ± 2σ	95.45%
μ ± 3σ	99.73%
μ ± 4σ	99.994%

Common Z values

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

Enter values or expressions: type directly into the Normal Distribution Calculator — keyboard friendly.
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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.

Related Guides

Source & Updates

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

Last updated: 2026-08-04

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

Cite this page: YiJianSuan. Normal Distribution Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/normal-distribution

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