Normal Distribution Calculator (PDF + CDF)
Compute probability density and cumulative probability at x for given μ, σ.
Result
P(X ≤ x)
0.747508
z = 0.6667 · PDF = 0.02129653.
- x110
- μ100
- σ15
- z-score0.666667
- PDF f(x)0.0212965337
- CDF P(X ≤ x)0.74750753
- P(X > x)0.25249247
- Percentile74.75%
Step-by-step
- z = (x − μ) / σ = (110 − 100) / 15 = 0.6667.
- PDF f(x) = (1 / (σ√(2π))) × e^(−z²/2) = 0.02129653.
- CDF Φ(z) ≈ 0.747508 (Abramowitz-Stegun approximation).
How to use this calculator
- Enter x.
- Enter μ and σ of the distribution.
- Read PDF + CDF + percentile.
About this calculator
The normal (Gaussian) distribution N(μ, σ²) is the bell curve. PDF f(x) = (1/(σ√(2π))) exp(−(x−μ)²/(2σ²)). CDF Φ(z) is the area to the left of z under the standard normal — the value used in p-values, confidence intervals, and percentiles. ~68% of values lie within 1σ of mean, 95% within 2σ, 99.7% within 3σ. The most fundamental distribution in statistics — central limit theorem makes sample means approximately normal.
Frequently asked
Standardized value: (x − μ) / σ. Tells you how many standard deviations x is from the mean.
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