  Let $a>1$. Denote by $l_a(p)$ the multiplicative order of $a$ modulo $p$. We
look for an estimate of sum of $\frac{l_a(p)}{p-1}$ over primes $p\leq x$ on
average. When we average over $a\leq N$, we observe a statistic of
$C\mathrm{Li}(x)$. P. J. Stephens ~\cite[Theorem 1]{S} proved this statistic
for $N>\exp(c_1\sqrt{\log x})$ for some positive constant $c_1$. Upon this
result, we give an explicit value of $c_1$. In fact, ~\cite[Theorem 1, 3]{S}
hold with $N>\exp(3.42\sqrt{\log x})$, and ~\cite[Theorem 2, 4]{S} hold with
$N>\exp(4.8365\sqrt{\log x})$. Also, we improve the range of $y$, from $y\geq
\exp((2+\epsilon)\sqrt{\log x\log\log x})$ in ~\cite[Theorem 1]{LP}, to $y >
\exp(3.42\sqrt{\log x})$.
