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

S_{h}(x) = ∑_{m=x+1}^{x+h} (\frac{m}{p}) ,

where p is a prime and (\frac{m}{p}) is the Legendre symbol, is normal as h, p → ∞ such that \frac{log h}{log p} → 0. We prove a similar result for sums of the form

S_{h}(x_{1}, …, x_{n}) = ∑_{z_{1}=x_{1}+1}^{x_{1}+h} ⋯ ∑_{z_{n}=x_{n}+1}^{x_{n}+h} (\frac{z_{1} + ⋯ + z_{n}}{p} ) .