In many applications, it is desirable to achieve a signal that is as close as possible to ideal white noise. One example is in the design of an artificial reverberator, whereby there is a need for its lossless prototype output from an impulse input to be perceptually white as much as possible. The Ljung-Box test, the Drouiche test, and the Wiener Entropy—also called the Spectral Flatness Measure—are three well-known methods for quantifying the similarity of a given signal to ideal white noise. In this paper, listening tests are conducted to measure the Just Noticeable Difference (JND) on the perception of white noise, which is the JND between ideal Gaussian white noise and noise with a specified deviation from the flat spectrum. This paper reports the JND values using one of these measures of whiteness, which is the Ljung-Box test. This paper finds considerable disagreement between the Ljung-Box test and the other two methods and shows that none of the methods is a significantly better predictor of listeners' perception of whiteness. This suggests a need for a whiteness test that is more closely correlated to human perception.