Let us consider a binary compound of type $AB_{n}$, where A and B both are metals. On normalization it can be written as $A_{x}B_{y}$, where x and y satisfy the following conditions:
\begin{equation}
x \leq y
\end{equation}
\begin{equation}
x + y = 1
\end{equation}
The weighted average of metallic electronegativity difference given by Villars, et.al is :
\begin{equation}
\Delta\chi = 2x(\chi_{A} - \chi_{B})
\end{equation}
where '$\chi_{A}$' and `$\chi_{B}$' are electronegativity of 'A' and 'B' in Martynov-Batsanov scale respectively.
Derivation:
The terms 'x' and 'y' are function of n and is given as:
\begin{equation}
x = \frac{1}{n+1}
\end{equation}
\begin{equation}
y = \frac{n}{n+1}
\end{equation}
where 'x' and 'y' satisfies the condition mentioned above.
In binary compound with chemical formula $AB_{n}$ we have 1 atom of A, n atoms of B and total of n+1 atoms.
1] The maximum number of pairs of A-B possible is : n
2] The total number of pairs possible with n+1 atoms is : $\binom{n+1}{2} = \frac{n(n+1)}{2}$
where, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
The weighted average of `s' is given as:
\begin{equation}
<s>_{w} = \frac{\Sigma_{i=1}^{t} w_{i}x_{i}}{\Sigma_{i=1}^{t} w_{i}}
\end{equation}
In our case, the weight for A-B ($w_{A-B}$) pairs is given as:
\begin{equation}
w_{A-B} = \frac{n}{\frac{n(n+1)}{2}} = \frac{2}{n+1}
\end{equation}
The weight for other possible pairs ($w_{other}$) is given as:
\begin{equation}
w_{other} = \frac{\binom{n}{2}}{\binom{n+1}{2}} = \left(\frac{\frac{n(n-1)}{2}}{\frac{n(n+1)}{2}}\right) = \frac{n-1}{n+1}
\end{equation}
Therefore the weighted average of metallic electronegativity difference is given as:
\begin{equation}
\Delta \chi = \frac{2}{n+1}(\chi_{A} - \chi_{B})= 2x(\chi_{A} - \chi_{B})
\end{equation}
