If some particles of liquid in a solution convert to vapour then why can't some liquid particles convert to solid? I was reading about the vaporisation of liquid when I came across the following statement.


  The average energy of the particles in a liquid is governed by the
  temperature. The higher the temperature, the higher the average
  energy. But within that average, some particles have energies higher
  than the average, and others have energies lower than the average.Some
  of the more energetic particles on the surface of the liquid can be
  moving fast enough to escape from the attractive forces holding the
  liquid together. They evaporate.


Now if some particles have energies lower than the average, then why don't they convert into solids in the same way the particles with higher energies convert to gases. 

If we cover a vessel filled with water with a plate, after some time, we observe droplets of water accumulated at the plate because of vaporisation of water. Why don't we observe any changes that show liquid particles converting to solid?
If water molecules go into the gas phase from the liquid then they are in a different phase. But water molecules go into the gas phase one at a time. If there are the same number of water molecules leaving the liquid as coming into it then the atmosphere above the liquid would have 100% humidity and be in equilibrium with the liquid. 

Now if some small number molecules of water happen to line up to form an ice crystal within the water itself what happens? Such an ice crystal is thermodynamically unstable and then "melts" within the liquid phase. Since the whole process takes place in the liquid phase it is virtually undetectable. 

Now you also have to realize that form ice crystals big enough to detect visually requires more than a handful of molecules. You'd need billions of billions of molecules. That is just a statistical impossibility.