  The signal to noise ratio efficiency $\epsilon_{\rm SNR}$ in axion dark
matter searches has been estimated using large-statistic simulation data
reflecting the background information and the expected axion signal power
obtained from a real experiment. This usually requires a lot of computing time
even with the assistance of powerful computing resources. Employing a
Savitzky-Golay filter for background subtraction, in this work, we estimated a
fully analytical $\epsilon_{\rm SNR}$ without relying on large-statistic
simulation data, but only with an arbitrary axion mass and the relevant signal
shape information. Hence, our work can provide $\epsilon_{\rm SNR}$ using
minimal computing time and resources prior to the acquisition of experimental
data, without the detailed information that has to be obtained from real
experiments. Axion haloscope searches have been observing the coincidence that
the frequency independent scale factor $\xi$ is approximately consistent with
the $\epsilon_{\rm SNR}$. This was confirmed analytically in this work, when
the window length of the Savitzky-Golay filter is reasonably wide enough, i.e.,
at least 5 times the signal window.
