Hallo Abhishek,

you are totally right! Extrapolation is only possible, if you have a really good idea of what is happening in the 'dead-zone'. That's the reason why I wrote all the time 'Assume, you SHURELY know, that...'

But even if you shurely know how your reference curve looks like, it's nevertheless a big problem to make an extrapolation basing on a data set which is not error free!

Situation looks a bit different, when only an interpolation has to be made. There are very powerful statistical tools, named chi-square distribution and F-test. This allows you to determine the goodness of a fit, and by this tells you of which order your power-series polynomial approximation PROBABLY is. But, of course, there's always some uncertainty. It's just the nature of statistics. And spoken in terms of statistics, the only thing you get when producing a set of data points (in the hope they are reference points) is always and only a GUESS of reference curve. And you should repeat this procedure of taking data points infinitely often, so that your guess becomes more and more trusty.

I can tell you how we dealt with taking data points at university. Assume you take the energy spectrum of a nuclear experiment and your data comes from a multichannel analyzer. Then you will have a multichannel pulse-height spectrum. You will see on it some Gaussian peaks plus a background. Interesting question is now, how many Gaussian peaks are there! Problem is, that the plot is very noisy of course, and you never shurely know, whether there is indeed a peak or just a peak looking noise figure. Then, when publishing your results while giving a talk, you don't state: 'Hey guys, there are three peaks!' But you state: 'If we assume, that there are two Gaussian peaks, F-test gives us the following result. But if we assume three peaks, F-test looks like ...'

Some physicists spend their whole life with hunting for a certain Gaussian peak... And some of them got the Nobel-price!

The only way to reduce the uncertainty is to increase the amount of data points, while repeating your experiment again and again, and, of course, by having this experiment repeated by many colleguae teams all over the world.

Kai