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Quantizing deep convolutional neural networks for image super-resolution substantially reduces their computational costs. However, existing works either suffer from a severe performance drop in ultra-low precision of 4 or lower bit-widths, or require a heavy fine-tuning process to recover the performance. To our knowledge, this vulnerability to low precisions relies on two statistical observations of feature map values. First, distribution of feature map values varies significantly per channel and per input image. Second, feature maps have outliers that can dominate the quantization error. Based on these observations, we propose a novel distribution-aware quantization scheme (DAQ) which facilitates accurate training-free quantization in ultra-low precision. A simple function of DAQ determines dynamic range of feature maps and weights with low computational burden. Furthermore, our method enables mixed-precision quantization by calculating the relative sensitivity of each channel, without any training process involved. Nonetheless, quantization-aware training is also applicable for auxiliary performance gain. Our new method outperforms recent training-free and even training-based quantization methods to the state-of-the-art image super-resolution networks in ultra-low precision.