Compression Spring Forces and Deflection

 F = k (L_f - L_d)

 or

L_d = L_f - (F / k)

Where;

F is the force applied to the compression spring (lbs)
k is the spring rate, often specified in catalogs. (lbs/inch) Also, see below for calculation of spring rate. 
L_f is the length of the spring with zero deflection, ie, no force applied. Often called "free length" (inches).
L_d is the length of the spring with the force F applied. (inches).

There is a comprehensive compression spring calculator/designer here at efunda.

Spring Rate (calculated)

k = (G d⁴) / (8 n D³)

Where;

k is the spring rate. (lbs/inch)
G is the modulus of rigidity (also called shear modulus) of the spring material (psi), See below. 
d is the spring wire diameter (inches) 
n is the number of active coils (less than the total number of coils) 
D is the mean spring coil diameter.(inches). = Spring dia. - wire dia.

G = E / (2 (1+v))

E is the elastic modulus of the spring material (psi).
v is the Poisson ratio of the material.