If $X$ is a real random variable with a pdf, variance/standard deviation is a measure of $X$'s dispersion about the pdf's central tendency, which in turn is referred to as the mean of $X$. For many, variance acts as a measure of risk and uncertainty.
Shannon entropy is a measure of the unpredictability of $X$, and is frequently compared with and does have mathematical links to variance. The estimated entropy of $X$'s pdf has been shown to scale alongside variance, in that, as $X$ is transformed by scaling it with increasing variance, entropy will correspondingly shift or grow in value, indicating more spread in the distribution like how the increasing variance indicates more dispersion. Transformations of the mean, however, will not impact the entropy rate.
What is the information analogue of the mean then?
If there is none, does this show a fundamental weakness in information theory? Is entropy and mutual information's inability to distinguish between negative and positive source data for its input probabilities make these measures blind/ignorant to 'the facts of the data', making them uninferrable with respect to one of the most important distributional concepts, central tendency/location?