Normal Distribution Calculator (PDF + CDF)

Compute probability density and cumulative probability at x for given μ, σ.

Result

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General calculation reads

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Scientific calculator (Casio fx-991EX)

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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

What's a "z-score"?+

Standardized value: (x − μ) / σ. Tells you how many standard deviations x is from the mean.

Why does normal distribution show up everywhere?+

Central limit theorem — sums or averages of many independent random variables converge to normal, regardless of underlying distribution.

How accurate is the CDF approximation?+

Abramowitz-Stegun rational approx: ~7 decimal digits. Plenty for typical statistical use.

68-95-99.7 rule?+

P(|X − μ| < σ) ≈ 0.68. P(|X − μ| < 2σ) ≈ 0.95. P(|X − μ| < 3σ) ≈ 0.997.

Discrete vs. continuous?+

Normal is continuous. P(X = exact value) = 0. Probabilities only meaningful for ranges.

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