Determining the empirical formula from mass % elements

The empirical formula is the simplest whole number ratio of atoms in a compound.  
"Empirical" means experimental.

Empirical formulae are determined by experiments that involve measuring the mass of elements (or compounds containing the elements).  
The experimental data may be expressed in mass percentage of the elements in the compound. 
Determining the empirical formula from these data is the reverse of the progression down the table above.
 
%mass
to ↓
 Any mass of H2O has 11.1 mass % H and 88.9 mass % O
mass in g 
to ↓
100 g H2O contains 11.1 g H and 88.9 g O
amount in moles
to ↓
100 g H2O contains 11.1 mol H and 5.55 mol O
mole ratio
to ↓
Any sample of water has
this ratio of
amount in moles of H to
amount in moles of O
2 mol H and 1 mol of O
formula   H2O

Note that if the ratio of amounts in moles does NOT come out to two whole numbers, both numbers in the ratio must be multiplied by the same factor to get the subscripts in the formula.
 
Using the mass % of N and O in N2O5 gives the ratio of the amount in moles n(N) to (n)O = 1 to 2.5.  Both would be multiplied by 2 to give n(N) to (n)O = 2 to 5