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I am hoping someone can recommend a useful statistical test. I have a set of data on an x-y plot which varies about the y=0 line in a seemingly random way. Each data point has a y error bar, which appears to be, in general, smaller than the standard deviation of the data.

I would like to apply a rigorous statistical test to calculate the probability that for the given individual one-sigma errors on each data point, the observed scatter about the y=0 line is plausible, or whether there is evidence for some kind of oscillation pattern?

A null hypothesis test was my first thought but this does not take into account the individual errors bars on the data points. Also I have noticed that many mathematicians don't hold the the null-hypothesis in much respect, nor does it offer a probability of the data points being randomly spread.

Thanks for any advise.

Natski