Following the standard conventions in the literature, the commutation relations of the Virasoro Lie algebra are given by
$$[L\_m,L\_n]=(m-n)L\_{m+n}+\delta\_{m,-n}\frac1{12}(m^3-m)c,$$
$$[c,L\_n]=0.$$


Similarly, following the standard conventions in the literature, the commutation relations of the Neveu-Schwarz super Lie algebra are given by
$$[L\_m,L\_n]=(m-n)L\_{m+n}+\delta\_{m,-n}\frac1{8}(m^3-m)c,$$
$$[J\_\alpha,J\_\beta]\_ +=2L\_{\alpha+\beta}+\delta\_{\alpha,-\beta}\frac12(\alpha^2-\frac14)c,$$
$$[L\_m,J\_\alpha]=(\frac12m-\alpha)J\_{m+\alpha},$$
$$[c,L\_n]=0,\qquad [c,J\_\alpha]=0.$$


The Virasoro algebra is a subalgebra of the Neveu-Schwarz algebra by $L\_n \mapsto L\_n$ and... $c \mapsto \frac32c$. Clearly, one could change the normalizations in the second set of formulas to avoid the annoyance of having to send $c$ to $\frac32c$.


So whyyy do people do it that way?  

Why do physicists take their standard formulas not consistent with each other?  

There must be a reason.



Why is there a discrepancy between the normalizations of the central terms for the commutation relations of the Virasoro versus Neveu-Schwarz Lie algebras?
As far as I can tell, a $\sigma$-model with $d$ dimensional target space will have Virasoro central charge $d$ with bosonic strings, and $3d/2$ with supersymmetric strings. I believe the normalizations were chosen so that the constant $c$ reflects the dimension of spacetime in which the strings are propagating (even if the specific model you are considering does not involve a spacetime manifold).


I don't have a good answer to your second question. Perhaps "it seemed like a good idea at the time".


