In this paper we consider the fractional type Marcinkiewicz integral operator \(\mu _{\Omega ,\beta }f(x) = \left( \int _{0}^{\infty } \left| \int _{\left| x-y \right| \le t } \frac{\Omega (x-y)}{\left| x-y \right| ^{n-1-\beta } } f(y)dy\right| ^{2}\frac{dt}{t^3} \right) ^{{1}/{2} },\quad 0<\beta <n,\) and the corresponding commutator \(\mu _{\Omega ,\beta }^b\) generated by \(\mu _{\Omega ,\beta }\) with \(b\in BMO(\mathbb {R}^n)\) . Typically, the bounds of \(\mu _{\Omega ,\beta }\) and \(\mu _{\Omega ,\beta }^b\) are \(\beta \) -dependent. We establish uniform quantitative weighted bounds for \(\mu _{\Omega ,\beta }\) and \(\mu _{\Omega ,\beta }^b\) with respect to \(\beta \) on weighted Lebesgue spaces. Moreover, the corresponding bounds for the classical Marcnkiewicz integral \(\mu _\Omega \) and the commutator \(\mu _\Omega ^b\) can be recovered from ones of \(\mu _{\Omega ,\beta }\) and \(\mu _{\Omega ,\beta }^b\) when \(\beta \rightarrow 0^+\) .
