Once measurements are taken, generally need to fit the data to find parameters such as critical exponents, using a multi-parameter fit to data which usually has large errors. This is very difficult!
For example requires a four parameter fit, and often have so much freedom that a wide range of values are possible. Usually need to assume , and get from other measurements.
Above formula is only true ``asymptotically close'' to . How close is that? Not always clear. May see ``apparent'' or pseudo-critical exponents, which change or disappear as the size of the system is increased and we move closer to .
There may also be corrections to the asymptotic formula, which are hard to distinguish from a small power-law exponent. In this case may need input from theory.