Noise that afflicts natural images, regardless of the source, generally disturbs the perception of image quality by introducing a high-frequency random element that, when severe, can mask image content. Except at very low levels, where it may play a purpose, it is annoying. There exist significant statistical differences between distortion-free natural images and noisy images that become evident upon comparing the empirical probability distribution histograms of their discrete wavelet transform (DWT) coefficients. The DWT coefficients of low- or no-noise natural images have leptokurtic, peaky distributions with heavy tails; while noisy images tend to be platykurtic with less peaky distributions and shallower tails. The sample kurtosis is a natural measure of the peakedness and tail weight of the distributions of random variables. Here, we study the efficacy of the sample kurtosis of image wavelet coefficients as a feature driving, an extreme learning machine which learns to map kurtosis values into perceptual quality scores. The model is trained and tested on five types of noisy images, including additive white Gaussian noise, additive Gaussian color noise, impulse noise, masked noise, and high-frequency noise from the LIVE, CSIQ, TID2008, and TID2013 image quality databases. The experimental results show that the trained model has better quality evaluation performance on noisy images than existing blind noise assessment models, while also outperforming general-purpose blind and full-reference image quality assessment methods.