Normal Distribution Calculator

Compute PDF, CDF, and inverse CDF for the normal (Gaussian) distribution with given μ and σ.

About Normal Distribution Calculator

The normal (Gaussian) distribution is the most important continuous distribution in statistics. It is completely defined by its mean μ and standard deviation σ. About 68% of values lie within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ.

Formulas

PDF

f(x) = (1/(σ√(2π))) × exp(−(x−μ)²/(2σ²))

μ: Mean
σ: Standard deviation
x: Point of evaluation

Standard Z

z = (x − μ) / σ

x: Value
μ: Mean
σ: Standard deviation

Worked example

With the default values below, Normal Distribution Calculator returns P(X ≤ x) of 0.97500217. Change any input above and every figure here is recalculated in your browser.

Inputs used

Mean (μ): 0
Standard Deviation (σ)minimum 0: 1
x value: 1.96

Results

P(X ≤ x): 0.97500217
PDF f(x): 0.05844094
P(X > x): 0.02499783
Mean: 0.0000
Variance: 1.0000

Step by step

1Values used
2PDF
3Standard Z
4P(X ≤ x)= 0.97500217
5PDF f(x)= 0.05844094
6P(X > x)= 0.02499783
7Mean= 0.0000
8Variance= 1.0000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is the standard normal distribution?

It is the normal distribution with μ = 0 and σ = 1. Any normal variable can be converted to standard normal using z = (x − μ)/σ.

Why is the normal distribution so important?

The Central Limit Theorem states that the sum (or mean) of many independent random variables tends toward a normal distribution regardless of the underlying distribution.