Let \(X_ 1,\dots,X_ n\) be i.i.d. rv's from a distribution function \(F\) with mean \(\mu\) and denote by \(\bar X\), \(\tilde X\), and \(s\), respectively, their mean, median and root mean square deviation. The authors consider the statistics \(M_ 1=n^{-1}\sum| X_ i-\bar X|\), \(M_ 2=n^{-1}\sum| X_ i-\tilde X|\), \(W_ i=M_ i/s\) \((i=1,2)\) and \(t_ i=\sqrt n(\bar X-\mu)/M_ i\) \((i=1,2)\) and derive asymptotic representations and Edgeworth expansions for them. They also consider the Gauss-Markov model and study the asymptotic distributions of statistics of the above forms depending on \(L_ 1\) and \(L_ 2\)-norms. \textit{H. J. Godwin} [Biometrika 33, 254-256 (1945)], \textit{R. C. Geary} [Biometrika 28, 295-305 (1936; Zbl 0015.40704)] and \textit{E. M. J. Herrey} [J. Am. Stat. Assoc. 60, 257-269 (1965)] considered \(M_ 1\), \(W_ 1\) and \(t_ 1\) when \(F\) is normal.