In this study, elastic response analyses of single degree of freedom (SDOF) structures under an extensive database of real earthquake excitation are investigated to clarify two particular issues: First is to determine whether the statistical distribution of ordered peaks in response displacement time histories follows Rayleigh distribution as it is a theoretical assumption valid for narrow band response of SDOF system with low damping value. Second is to look for a functional form to define amplitudes of the peaks in response displacement time histories as a function of structural period and response spectral displacement (Sd). Probability density functions (PDFs) were calculated for structural periods between 0.1-5 sec. using order statistics approach. Numerical results support that Rayleigh distribution describes the probabilistic characteristics of ordered peaks in elastic earthquake response adequately. Scale parameters of the distribution were calculated. It was observed that there is a clear logarithmic relationship between the peak number and the normalized peaks’ amplitudes in displacement time histories. Hence, a nonlinear regression model was proposed to define the peaks’ amplitudes in response displacement time histories of a SDOF system with known response Sd values and structural periods. In another words the model is capable of estimating number of times of a Sd that can be repeated by a structure without any damage. It is observed that even in the elastic systems, response values that are quite close to the value of the highest peak displacement are repeated many times. In general, for SDOF systems having natural period of vibration greater than or equal to 0.9 sec., oscillation having 90 % of the value of highest peak is repeated two times under the earthquake excitation. SDOF systems having natural period of vibration vary between 1 and 1.7 sec., ratio between second largest peak and Sd is about 95 %.Those calculations are essential for determining start of low-cycle fatigue behavior particularly for design of column-beam joints of steel structures or bridges.