Davenport and Erdos showed that the distribution of values of sums of the form

$$S_h(x)=\sum_{m=x+1}^{x+h} \left(\frac mp \right),$$

where $p$ is a prime and $\left(\frac mp \right)$ is the Legendre symbol, is normal as $h, p\to\infty$ such that $\frac{\log h}{\log p}\to 0$. We prove a similar result for sums of the form

$$S_h(x_1, \ldots, x_n)=\sum_{z_1=x_1+1}^{x_1+h} \cdots \sum_{z_n=x_n+1}^{x_n+h} \left( \frac {z_1+ \cdots+z_n}{p} \right).$$