Inverse Chi Squared Calculator - Calculator Academy

Enter the degrees of freedom (df) and the chi-squared statistic (χ²) into the calculator to determine df/χ² (the inverse of the reduced chi-squared, χ²/df).

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Inverse (df/χ²) Formula

The following formula is used to calculate df/χ² (the inverse of the reduced chi-squared, χ²/df) for a given degrees of freedom and chi-squared statistic.

I = \frac{df}{\chi^2}

Variables:

I is the inverse ratio (df/χ²), i.e., the reciprocal of the reduced chi-squared (χ²/df)
df is the degrees of freedom
χ² is the chi-squared statistic

To calculate df/χ², divide the degrees of freedom by the chi-squared value.

What is df/χ²?

On this page, “inverse chi-squared” refers to the simple ratio df/χ², which is the reciprocal of the reduced chi-squared statistic (χ²/df). It is a unitless summary sometimes used when assessing goodness of fit: values near 1 indicate χ² is close to df; values greater than 1 occur when χ² < df; and values less than 1 occur when χ² > df (often interpreted alongside measurement uncertainties and model assumptions). This is different from the inverse-chi-squared (or scaled inverse-chi-squared) probability distribution used in Bayesian statistics, and also different from the inverse chi-squared CDF (the chi-squared quantile function).

How to Calculate df/χ²?

The following steps outline how to calculate df/χ².

First, determine the degrees of freedom (df).
Next, determine the chi-squared value (χ²).
Finally, calculate the inverse ratio using the formula I = df / χ².
After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem :

Use the following variables as an example problem to test your knowledge.

Degrees of freedom (df) = 10

Chi-squared value (χ²) = 5

Inverse (df/χ²) = 10 / 5 = 2