Rubber band elongates like s-curve A normal rubber band (brownish yellow) with about 1 mm^2 cross section and approximate slack length of 170 mm is suspended vertically and gradually loaded with a number of weights (each weighing 9.36 grams) and the resulting elongation of the eleastic band is measured. The results are as shown here
The early measurement shows a slightly "weaker" band than the later measurement. I suspect the temperature of the band might have been higher at the time of the early measurement than it was later, so this could be an explanation. Notice the s-shape of both curves indicating a substantial deviation from Hookes law and requiring at least a third order approximation as shown in the following 2 charts

What I really want to ask is if anybody can explain the s-curved shape of the elongation versus load as this seems contrary to my intuition.
 A: Your elastic band is being stretched a LOT: starting with a length of 170 mm, you are increasing its length by about 3x (elongation around 340 mm).
So first of all, your band is becoming thinner as it stretches (as given by the Poisson ratio of the material). This means that for a given force, the stress (which is force per unit area) increases faster than you were expecting. This gives rise to the initial rising of the curve: while the Youngs modulus is stress divided by strain, and your stress is bigger than indicated by the force, you will see an increase in slope.
Eventually you get to the point where the molecules in the elastic band are almost completely straightened out. Where the initial elasticity was due to the fact that you were straightening "slightly bent" polymers, once they are straight, the force you need to further extend them is different: you are starting to stretch atomic bonds, move straight molecules past each other, ...
In short - the elastic mechanism has changed. And with it, the Young's modulus. Which is why you get less extension per added force. You have reached the elastic limit of the material.
There is a lot of material online discussing this. See for example this lecture, from which I quote:

Releasing the stress applied to a cross-linked rubber, leads to the coiling
  back of extended chains, since a retraction allows the chains to adopt
  higher entropy conformations.

This "recoiling" is the thing I was talking about: as the chains straighten out, their geometry is sufficiently different that their elastic properties change. A diagram from the same lecture bears this out, and shows your data is sensible:

