This is a nice example of the difference between Physics and Mathematics.
The mathematical model says that, for a limited range of elongations, modulus of the force and displacements are proportional. However, real measurements deviate from the exact Hooke's law for two reasons.
The first is due to the unavoidable perturbations connected to each measurement. The exact sources of such perturbations may be difficult to identify uniquely and depend on the details of the protocol, on the hardware, on the external conditions, just to mention a few causes. Usually they introduce a kind of random variation of the individual measurements which can be analyzed and controlled with statistical methods and this is what people call the statistical error. In principle, it can be systematically reduced by increasing the number of measurements.
A second class of deviations originates from non-random sources. Collectively, and on the base of the effect on results, this class of deviation is called systematic error. Once again there are many different possible causes of systematic error. In the case of a spring constant measurement I would list:
- bad calibration of the instruments;
- action of a physical mechanism not present in the model like, for example, the presence of a non-linear regime or a dependence of the force constant on temperature with experiments performed in conditions of systematic increase or decrease of temperature;
- experimental points coming from completely different experiments and measurement methods.
Therefore, you see that the world, out of the textbooks is quite complex. The fascinating thing is that, notwithstanding such difficulties, it is still possible to reduce all the sources of uncertainty and to build predictive and accurate theories out of noisy experimental data.
The typical problem of the experimental data is how to be able to assign a probability that a given set of noisy data is compatible with a physical hypothesis.
I assume that the problem, with its quite large deviation from linearity, was proposed as an exercise related to error analysis.