What does the standard deviation tell us about the data that in not obvious
from the mean?

Explanation:

One liner: Its a measure of how much close to the mean value the actual data points are.

Consider you have ten people and you are given that their mean age is 30.

Why are you given the average age and not the age of each person separately? for easiness of analysis. They don't want you to stare blankly at 10 different values. When we try to represent the information contained in ten values by a single value, of course, there is a trade off. You lose the accuracy of the information. So, there is a need to find out how good is this 'easy representation'.

Lets take the case. Average age is of 10 persons is 30. There can be n combinations of their ages. they could be i) 1,1,1,1,1,59,59,59,59,59 or they could be ii)30,30,30,30,30,30,30,30,30,30 or they could be ii)15,15,15,15,15,45,45,45,45,45.

In the first case, the actual values are spread far away from the mean. In the second case, all of them are exactly on mean and in the third case, the values are moderately separated away. How to distinguish between these? Here comes the need for standard deviation. A high standard deviation signifies high deviation of data points from the mean. A moderate SD signifies a moderate deviation of data points from the mean and a low (can be even zero) signifies that the data points are close to the mean.

Adding one more quantity of information, you get to have a better idea of the data set (rather than just having the mean). But then, the trade off is, you need to carry more information.