It seems you have a common misconception about how bone deformation weighting is calculated. The way it actually works is as follows:
If a vertex belongs to one or more vertex groups which are parented to bones, that vertex's bone-deformed location will be equally split between the bones' locations according to the ratio of their respective weights.
The key here is that deformation is calculated based on the ratio of weights per vertex, not on the absolute value of each weight. So if a vertex is only parented to one bone it's deformation location will be entirely decided by that one bone, no matter how small a weight it has in that bone's vertex group.
To get the vertex's position to be 10% affected by the bone you have to specify what affects the other 90%. You can do this by adding another bone that does not move to your armature. Then add a new vertex group for this bone and assign the vertices to this vertex group with a weight of
In the above example both "legs" are parented to
jointBone with weight
0.1, but the right one is also parented to
basisBone with weight