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I have created a histogram of velocities for thousands of moving objects. I have bin sizes of 1 based on object weight. So bins 1-20, for weight 1gram to 20 grams. So that's the "x-axis".

The y-axis, or height of the bins is based on averages of velocities for these objects. I take all the 1 gram objects and find their average velocity, and that is the height of the 1 gram bin. Then the same for all the 2 gram bin, find their average velocity and plot that as the bin height.

My question is, how can I find if there is a significance between the bins. First of all, finding significance between 20 bins is a LOT of comparisons, is there an easy way to do that?

But even if I can just compare 2 or 3 at a time that would be useful. So I can say the 1 gram objects have a significantly different velocity than the 2 gram objects.

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    $\begingroup$ If you have a lot of data, you should avoid binning them. Consider using a non-parametric smoother to show the bivariate relation between these data. And consider also that tons of data circumvents the need for traditional testing: you are getting closer to simply showing people what your study is capable of producing with descriptive statistics. It is more powerful than tests. $\endgroup$
    – AdamO
    Commented Nov 18, 2016 at 18:27

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By significance I assume you mean significantly different velocities. You might try a kernel smooth as suggested by @AdamO . In addition, one can plot confidence intervals for the appropriate location measurement of the y-axis data, that may be easier to understand than probabilities. And, the confidence interval ranges will obviously be a function of the smoothing width or histogram category widths. When these intervals do not overlap, the differences would be significant.

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  • $\begingroup$ Yes, significantly different velocities is what I'm looking for. Actually I expect them to be the same, and most of my bin heights are about the same height, like a flat line across all 20 bins tops. But I'd like to be able to say for sure that there is no difference between them, to quantify that in some way. I've never even heard of a kernel smooth, but I'm going to look in to it and see what I come up with. It looks like its basically a regression line or something that I would plot for all the averages without binning. $\endgroup$
    – Nertskull
    Commented Nov 18, 2016 at 19:44
  • $\begingroup$ reference.wolfram.com/language/ref/… $\endgroup$
    – Carl
    Commented Nov 18, 2016 at 19:54
  • $\begingroup$ Actual data as events are hypersparse, i.e., nothing, nothing, nothing, event, nothing, nothing,,,would be an exaggerated density. To form an image some type of smoothing is needed. Binning is, unfortunately, not usually as stationary as it could be, and smoothing without binning can give better x-axis location, and is also not as 'step function like' as binning. $\endgroup$
    – Carl
    Commented Nov 18, 2016 at 22:28

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