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Normal Distribution Calculator

Online normal distribution: standardize with Z=(X−μ)/σ and find the cumulative probability Φ(Z).

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Z=(X−μ)/σ
Mean
σ>0
Value

📖 Tutorial | Normal Distribution

1. Definition
For X~N(μ,σ²), standardize with Z=(X−μ)/σ, so Z~N(0,1) with CDF Φ(Z).
2. Formula & Notation
μ,σMean, std dev
3. Properties & Laws
  • Symmetric bell shape;
  • 68-95-99.7% within ±1,2,3σ;
  • σ>0;
  • Φ(−z)=1−Φ(z).
4. Steps
  1. Enter μ,σ,x;
  2. Click "Calculate";
  3. Or click "Load Example".
5. Worked Examples
Example: standard normal x=1.96: Z=1.96, Φ≈0.975.
6. Common Pitfalls
Divide by σ;
σ>0;
Φ is the left-tail CDF.

❓ FAQ | Normal Distribution

Standardize?
(x−μ)/σ.
Φ?
CDF.
68-95?
±σ.
Symmetric?
About μ.
Example?
0.975.
Two-sided 95?
±1.96.