Let $G$ be a connected graph on $n$ vertices. The thorn graph $G^\star$ of $G$ is obtained from $G$ by attaching to its $i$-th vertex $p_i$ new vertices of degree one, $p_i \geq 0$, $i=1,2,\ldots,n$. Let $d(G)$ be the sum of distances of all pairs of vertices of $G$. We establish relations between $d(G)$ and $d(G^\star)$ and examine several special cases of this result. In particular, if $p_i=\gamma-\gamma_i$, where $\gamma$ is a constant and $\gamma_i$ the degree of the $i$-th vertex in $G$, and if $G$ is a tree, then there is a linear relation between $d(G^\star)$ and $d(G)$, namely $d(G^\star)=(\gamma-1)^2\,d(G)+[(\gamma-1)n+1]^2$.