Inverse Normal Distribution: Definition & Example


The term inverse normal distribution refers to the method of using a known probability to find the corresponding z-critical value in a normal distribution.

This is not to be confused with the Inverse Gaussian distribution, which is a continuous probability distribution.

This tutorial provides several examples of how to use the inverse normal distribution in different statistical softwares.

Inverse Normal Distribution on a TI-83 or TI-84 Calculator

You’re most likely to encounter the term “inverse normal distribution” on a TI-83 or TI-84 calculator, which uses the following function to find the z-critical value that corresponds to a certain probability:

invNorm(probability, μ, σ)

where:

  • probability: the significance level
  • μ: population mean
  • σ: population standard deviation

You can access this function on a TI-84 calculator by pressing 2nd and then pressing vars. This will take you to a DISTR screen where you can then use invNorm():

invNorm function on a TI-84 calculator

For example, we can use this function to find the z-critical value that corresponds to a probability value of 0.05:

Z critical value for left-tailed test on a TI-84 calculator

The z-critical value that corresponds to a probability value of 0.05 is -1.64485.

Inverse Normal Distribution in Excel

To find the z-critical value associated with a certain probability value in Excel, we can use the INVNORM() function, which uses the following syntax:

INVNORM(p, mean, sd)

where:

  • p: the significance level
  • mean: population mean
  • sd: population standard deviation

For example, we can use this function to find the z-critical value that corresponds to a probability value of 0.05:

The z-critical value that corresponds to a probability value of 0.05 is -1.64485.

Inverse Normal Distribution in R

To find the z-critical value associated with a certain probability value in R, we can use the qnorm() function, which uses the following syntax:

qnorm(p, mean, sd)

where:

  • p: the significance level
  • mean: population mean
  • sd: population standard deviation

For example, we can use this function to find the z-critical value that corresponds to a probability value of 0.05:

qnorm(p=.05, mean=0, sd=1)

[1] -1.644854

Once again, the z-critical value that corresponds to a probability value of 0.05 is -1.64485.

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