What kind of a wall?
A cave wall? If yes, it depends on the layer you're in.
When mining, certain rock walls are harder than others. If you attempt to mine a wall that is too hard for you, you will get a message along the lines of "This wall is slightly/far/much too hard for you to mine". The hardness of a wall appears to be equal to the quality of ore and stone you receive from mining it. In order to mine a wall, you appear to need sqrt(Strength * Tool_Quality) >= Wall_Hardness. For example, in order to mine a quality 10 rock wall with a quality 10 stone axe or pickaxe, you would need 10 strength. Note that the pickaxe does not appear to be any stronger than a stone axe at mining, merely faster.
Source:
http://ringofbrodgar.com/wiki/MiningLayer 1 is an average of q10, meaning you'd need a square root(strength * tool quality) of 10 or higher.
Since the lowest quality of a stone axe is 5, and strength is started at 10, you should be able to mine quality 7 or below without ANY upping of stats.