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For more on marking an answer as the "Best Answer", please visit our FAQ.The formula is correct and you should obtain a straight line graph (allowing for experimental error), which does not pass through the origin, when you plot a graph of T^2 (s^2)) against m(kg). If you find the gradient of the best fit line then the elastic constant k(N/m) can be found by multiplying the reciprocal of this gradient by 4Pi^2. A second graph is not required.
You can compare this result with the value of k obtained by plotting a graph of F(N) where F=mg against the maximum deflection x(m) where k= gradient of this graph.
All this would have to be carried out for each material.
The elastic constant does not vary with mass it is constant until the physical dimensions and/or the material is changed.