Q. How much information do I have about variability in this population?
A: None. I have just one observation and so I have no information about variability.
The first observation tells me nothing about variability.
It's not until I measure the bodyweight of a second person that I get any information about variability.
If the second person has a bodyweight of 62kg, I'm thinking there isn't much variability. If my second person weighs 80kg, then I'm thinking there may be a lot of variability in this sample.
Degrees of freedom tell us how much information we have about variability.
The first observation tells us nothing about variability.
See http://onlinelibrary.wiley.com/doi/10.1002/pst.27/epdf for a more complete explanation of degrees of freedom and extension to analysis of variance and other examples.