Is spring constant really a constant value? ( Assume the spring is not changed ) l just encountered a problem that is about a string in harmonic motion. The question states that the cord is elastic and gives a table like this

The question didn't states that the cord changes, only the mass and the length are the variables, however, l found out that  for each Weight, the value of string constant is slightly different from each other, why? Isn't k only depends on the material of the spring, if the spring did not change, k should be constant all the time.
If spring constant is not a constant value, why it is called spring constant
What caused k varies?
 A: Yes, the force a spring (or generally any material obeying Hooke's law) exerts varies linearly with the elongation, as long as it is sufficiently small . Looking at a typical stress/strain diagram, for materials it is accepted that Hooke's law becomes invalid around a strain of 1%. The reason your constant varies is because of supposed measurement errors though.
A: 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.
A: Under the elastic limit, an object under tension experiences an increment/decrement of length $\Delta x$ (with initial length being $x_0$)and this causes a restoring force,$F$, to act, which is given by:
$$F = \left ( \frac {YA}{x_0} \right)\Delta x = k\Delta x$$
$$\Rightarrow k=\frac {YA}{x_0}$$
Here, $Y$ is the Young's modulus of elasticity (a constant for a given material), and $A$ is the cross sectional area of the material. 
Clearly, you can see that $k$ varies from one material to another. Also it varies from one object to other (for the same material).
