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Formula verified · Deterministic calculation

Normal Distribution Calculator

Enter x, the distribution mean and a positive standard deviation to evaluate the normal density and approximate cumulative probability.

P(X ≤ x)0.5000000005
Z-score0
Probability density0.398942280401
Upper-tail probability0.4999999995
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Result methodology

The certified engine standardizes x, evaluates normal density and uses a deterministic error-function approximation for cumulative probability.

Solution steps
  1. z = (x-μ)/σ = 0
  2. PDF = 0.3989422804014327
  3. P(X≤x) ≈ 0.5000000005

How this calculator works

The value is standardized with z=(x−μ)/σ. Density uses the normal PDF; cumulative probability uses a deterministic error-function approximation.

Formula

z=(x−μ)/σ; f(x)=exp(−z²/2)/(σ√(2π))

Standardizes x and evaluates normal density plus an approximation to the cumulative distribution.

Worked example

At the mean is a verified test case used by the calculation engine.

Assumptions and limitations

  • Standard deviation must be positive.
  • The modeled variable is normally distributed.
  • The cumulative probability is a numerical approximation.

Methodology & sources

This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint.

  • CalcuMint deterministic statistics engine

Frequently asked questions

What is the CDF result?

It is the approximate probability P(X≤x) under the specified normal distribution.

What is the upper-tail probability?

It is 1−CDF, the approximate probability above x.

Related calculators

Z-Score CalculatorStandardize a value using a supplied mean and positive standard deviation.Standard Deviation CalculatorCalculate population or sample standard deviation and variance.Binomial Probability CalculatorCalculate the probability of exactly k successes in n independent trials.