  To solve the obscureness in measurement brought about from the weak
ergodicity breaking appeared in anomalous diffusions we have suggested the
time-averaged mean squared displacement (MSD) $\bar{\delta^2 (\tau)}_\tau$ with
a integral interval depending linearly on the lag time $\tau$. For the
continuous time random walk describing a subdiffusive behavior, we have found
that $\bar{\delta^2 (\tau)}_\tau \sim \tau^\gamma$ like that of the
ensemble-averaged MSD, which makes it be possible to measure the proper
exponent values through time-average in experiments like a single molecule
tracking. Also we have found that it is originated from the scaling nature of
the MSD at a aging time in anomalous diffusion and confirmed them through
numerical results of the other microscopic non-Markovian model showing
subdiffusions and superdiffusions with the origin of memory enhancement.
