Let \(X\) and \(Y\) be real vector spaces. One of main theorems of this paper states that a function \(f: X\to Y\) satisfies the functional equation \[ n^2{n-2\choose k-2}\, f\Biggl({x_1+\cdots+ x_n\over n}\Biggr)+ {n-2\choose k-1}\,\sum^n_{i=1} f(x_i)= k^2 \sum_{1\leq i_1<\cdots< i_k\leq n} f\Biggl({x_{i_1}+\cdots+ x_{i_k}\over k}\Biggr) \] if and only if \(f\) has the form \(f(x)= Q(x)+ A(x)+ f(0)\) for some additive function \(A\) and some quadratic function \(Q\), where \(k\) and \(n\) are integers with \(1< k< n\). Moreover, the author proves the Hyers-Ulam-Rassias stability of the above equation: Let \(X\) and \(Y\) be a real normed space and a real Banach space, respectively. If a function \(f: X\to Y\) satisfies \[ \begin{multlined} \Biggl\| n^2{n-2\choose k-2}\, f\Biggl({x_1+\cdots+ x_n\over n}\Biggr)+ {n-2\choose k-2}\, \sum^n_{i=1} f(x_i)\\ - k^2\sum_{1\leq i_1<\cdots< i_k\leq n} f\Biggl({x_{i_1}+\cdots+ x_{i_k}\over k}\Biggr)\Biggr\|\leq \theta \sum^n_{i=1}\| x_i\|^p\end{multlined} \] for all \(x_1,\dots, x_n\in X\) and for some \(\theta\in [0,\infty)\) and \(p\in (0,\infty)- \{1,2\}\), then there exists a unique quadratic function \(Q: X\to Y\) and a unique additive function \(A:X\to Y\) such that \[ \| f(x)- Q(x)- A(x)- f(0)\|\leq \varepsilon_1(x)+ \varepsilon_2(x) \] for all \(x\in X\), where \[ \varepsilon_1(x)= {k^p\theta\over {n-2\choose k-1}| k^2- k^p|} \| x\|^p\quad \text{and} \quad\varepsilon_2(x)= {(2^p+ 2)k^{p-2}\theta\over {n-3\choose k-2}| 2- 2^p|}\| x\|^p. \]