This paper introduces a scaled IIR filter based on stochastic computation. The stochastic IIR filter can provide an area-efficient hardware implementation that replaces a multiplier used in a traditional implementation by a simple logic gate. However, it strongly suffers from overflow of internal values as stochastic computation represents limited real values within -1 to 1, which significantly degrades the performance of the stochastic IIR filter. In order to maintain internal values within -1 to 1, the proposed stochastic IIR filter exploits a scaling method based on an L∞ norm. An input signal is scaled down by a scaling coefficient and then is scaled up after a feedback-loop block to provide a signal amplitude desired. As a design example, second-order low-pass IIR filters based on stochastic computation are designed and simulated in MATLAB. The proposed scaled stochastic IIR filter provides a similar response to an ideal floating-point IIR filter while a stochastic IIR filter without scaling degrades a signal amplitude by 19.2 dB with a frequency lower than a desired cutoff frequency.