Function Repository Resource:

# SigmaConfidenceLevel

Calculate the confidence level corresponding to width around the mean of a normal distribution

Contributed by: Julien Kluge
 ResourceFunction["SigmaConfidenceLevel"][σ] calculates the fraction of the normal distribution that falls within σ standard deviations of the mean.

## Details and Options

ResourceFunction["SigmaConfidenceLevel"] evaluates to exact outputs for exact inputs and certain values.
The associated one-sided p-value can be calculated by using p=(1-ResourceFunction["SigmaConfidenceLevel"][σ])/2.

## Examples

### Basic Examples (2)

Calculate the confidence level at one standard deviation:

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Calculate the p-value associated with 3σ:

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### Scope (5)

SigmaConfidenceLevel returns exact outputs for exact inputs:

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In physics, a discovery typically has to pass a five-σ of certainty test. Calculate the probability that a five-σ result is not a statistical fluctuation:

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A p<0.05 test corresponds to a maximum displacement from mean of about 1.64 standard deviations:

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SigmaConfidenceLevel can be defined as the integral between -σ and σ of the NormalDistribution with mean 0 and standard deviation 1:

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### Applications (1)

Fit the mean prediction bands of a linear model fit with a confidence level of 1σ, 2σ and 3σ:

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### Properties and Relations (1)

SigmaConfidenceLevel handles infinities:

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### Neat Examples (1)

The 5σ classification limit in physics for discoveries has a false-positive probability of approximately:

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Julien Kluge

## Version History

• 1.0.0 – 23 April 2020

## Author Notes

Julien Kluge Quantum Optical Metrology; Joint Lab Integrated Quantum Sensors Department of Physics Humboldt-Universität zu Berlin (julien@physik.hu-berlin.de)