Normal Distribution
For a normal distribution N(μ, σ²) with a given mean and standard deviation, compute the probability density (PDF), cumulative probability (CDF), upper-tail probability and standardized z value at any x.
Input
Enter the mean μ, the standard deviation σ and an evaluation point x to compute the probability density, lower-tail probability, upper-tail probability and standardized z for the normal distribution N(μ, σ²).
The value x to evaluate the probability at
The mean at the center of the distribution
The spread of the distribution (positive value)
Result
Lower-tail probability (CDF) up to x = 1
0.84134475
Upper-tail probability
0.15865525
Probability density f(x)
0.24197072
Standardized z
1
Probability density function (PDF) and lower-tail area
Cumulative distribution function (CDF)
How it works
- The normal probability density is f(x) = exp(-(x - μ)² / (2σ²)) / (σ √(2π)), where μ is the mean and σ is the standard deviation.
- The lower-tail probability (cumulative distribution function CDF) gives P(X ≤ x) and is computed as Φ(z) = (1 + erf(z / √2)) / 2, where z = (x - μ) / σ is the standardized value.
- The upper-tail probability is 1 minus the lower-tail probability and gives the probability that X is at or above x.
- The error function erf is evaluated to high precision with a combination of a series expansion and a continued fraction, so the result does not depend on approximate lookup tables.
- The standard deviation σ must be positive. A value of 0 or below leaves the distribution undefined.
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Normal Distribution