What does the “reduced” (χ^2) distribution look like for a particular “degree of freedom.”

First of all, let’s look at the chi-square function as a function of (χ^2) and the degrees of freedom (ν).

PDF[ChiSquareDistribution[ν],x]

Plot the chi-square function for ν = 30 degrees of freedom.

Plot[PDF[ChiSquareDistribution[30],x],{x,0,50},PlotRange->All,AxesOrigin->{0,0}]

Now we want to create a “normalized” reduced chi-square function  (Subscript[χ, red]^2) that we will call f[y,ν].  Currently, Mathematica does not have this function, so, we will create it.

Use the “Faltung” integral (a.k.a. the “folding” integral or the “convolution” integral, etc.) to scale the chi-square axis by a factor of ν, the degrees of freedom.

In the following transformation, let       x = (χ^2)       and        y = (Subscript[χ, red]^2)         and      ν = # of degrees of freedom.

f=.

f[y_,ν_]:=(∫_(-∞)^∞)DiracDelta[y-x/ν] PDF[ChiSquareDistribution[ν],x]\[DifferentialD]x

After performing the transformation, what is the expression for the reduced chi-square function for “ν” degrees of freedom?

Assuming[{ν>0&&χ2r>0},f[χ2r,ν]]

Plot the reduced chi-square function for an arbitrary number of degrees of freedom, ν.  In this case, ν=3.

Plot[Evaluate[f[χ2r,ν]/.ν->3],{χ2r,0,3},PlotRange->Automatic]

Notice that there is a broad range of “acceptable” reduced chi-squares that one can have for a small number of degrees of freedom.
Plot your own reduced chi-square function by substituting your own number of degrees of freedom in the equation above.

Also notice that the reduced chi-square distribution is normalized because both the “folding” function (i.e., the Dirac δ-function) and the chi-square function are normalized.  This is true for all degrees of freedom, ν ≥ 1.  For example,

Assuming[ν>0,(∫_0^∞)f[χ2,ν]\[DifferentialD]χ2]

Plot the reduced chi-square function for 30 degrees of freedom

Plot[Evaluate[f[χ2,ν]/.ν->30],{χ2,0,3},PlotRange->All]

Notice that when the degrees of freedom increases, the reduced chi-square function peaks about 1.0 and its width becomes narrower.  Try 300 degrees of freedom, and watch the width become even narrower; however, the mean stays centered around 1.00.

Let’s take a look at the “reduced” chi-square for 3 different degrees of freedom, ν = 2, 20, and 200.

Plot[Evaluate@Table[f[χ2r,ν],{ν,{2,20,200}}],{χ2r,0.,3.0},PlotRange->All,Filling->Axis]

Notice that the range of “acceptable” reduced chi-squares (e.g., 90% C.L.) becomes smaller as the number of degrees of freedom increases.