In this paper, we take up Taylor means to study the degree of approximation of $f\in L_p$ ($p\ge1$) under the $L_p$-norm and obtain a general theorem which is used to obtain four more theorems that improve some earlier results obtained by Mohapatra, Holland and Sahney [J. Approx. Theory 45 (1985), 363-374]. One of our theorems provides the Jackson order as the degree of approximation for a subspace of $Lip(\a,p)$ ($0<\a<1$, $p\ge1$) and generalizes a result due to Chui and Holland [J. Approx. Theory 39 (1983), 24-38].