How does the variance measure the information about the data? Variance measures the amount of information in the data set. How?
Mathematically, the variance is the measure of the variability in the data. So, how this can be explained as a measure of the amount of information in the data? Could you please explain this with a simple example?
I read the following post, here, which makes me confused about, how the variance measures the information about the data? I thought it is just a measure of how the point is far away from the mean.

Others already answered conveying perfectly some concepts but the major misunderstatement remains to be clarified: variance IS NOT information, is not even vaguely linked to information. In some application as PCA it is a parameter that allows you to evaluate how much information you can extract from your data with that specific tool, but this is not a measure of information of the data at all. I will not go into mathematical explanation because it is probably better to give basic ideas first (and there are very good formal explanation out there): information is the reduction of uncertainty, lack of variance in data does not correlate in any way with the information, it just means that the phenomenon is steady and this is a big amount of information because it reduces the a priori uncertainty about that phenomenon. The key concept is that information depends on the relation between the observer and the data, not on the data itself: if you think to source coding (data compression as an example) small variance means that you can use less symbols to encode the data and so it means less information, but if you think to channel coding (a repetition code for example) in a large set of "1" there could be no "useless" symbol because every symbol carries part of the original information, there are codes way better than the repetion code to increase the amount of information carried by each symbol, still it is the proof that having n+1 equal values conveys more information than n equal values and so even with 0 variance the information increases. Example: you receive a boolean information on a noisy channel with 0.5 cross-channel probability, if you receive "1" you have exactly 0 bit of information because there is no way to know if the sended bit was 0 or 1, if the sender repeats his message 3 times and you receive "111" you know that the probability that the message was 1 is 0.875 and with 0 variance you have a lot of information, passing from no information at all (1) to very good confidence (111).
