I'm preparing a presentation about parallel statistics. I plan to illustrate the formulas for distributed computation of the mean and variance with examples involving center of gravity and moment of inertia. I'm wondering if there is a physical interpretation of the third or higher central moments that I can use to help illustrate the general formula.
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$\begingroup$ That relates the properties of moments to graphs of PDFs, but is not useful for visualizing computations done on moments. I'd like to be able to, for example, write Welford's algorithm (en.wikipedia.org/wiki/…) on the board, and then draw a picture and say "This is $x_n-\overline{x_n}$" and have the equality become somewhat obvious. Also at that link is a streaming algorithm for the mean. It takes a few seconds to see why the formula works, but becomes immediately obvious if I draw it as a center of gravity problem. $\endgroup$– James KoppelJul 11, 2011 at 22:00
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4$\begingroup$ If you must preserve the interpretation of $x$ as a length, then anything above third moments will require too many dimensions to illustrate! In spirit, your question and comment are like the Ancient Greek drive to cast all arithmetic in terms of geometry. That indeed provides insight, but it also limits the scope of what can be done to a small number of dimensions (0 through 3). Creative use of graphics can help us visualize higher dimensions (and therefore, perhaps with higher moments). Thus I think you shouldn't dismiss graph-based illustrations out of hand. $\endgroup$– whuber ♦Jul 11, 2011 at 22:16
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$\begingroup$ I'm not sure what you mean by physical interpretation. (Perhaps it's that I don't think about mean or variance in terms of physical quantities). If you're just trying to to reprise the Welford algorithm, is it mathematically anything more than the insight that to compute statistics of the form $T = \frac{1}{N} \sum_{i=1}^n f(x_i)$, you can always compute them recursively via: $\;T_1 = f(x_1)$, and $T_i = ((N-1)\cdot T_{i-1} + x_i) / N \;\;$ ? Raw moments can always be computed this way, and formulas for centered moments can (presumably) be derived from these? $\endgroup$– jpillowJul 12, 2011 at 5:23
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2$\begingroup$ Closely related: stats.stackexchange.com/questions/132914, stats.stackexchange.com/questions/17595. $\endgroup$– whuber ♦May 8, 2015 at 18:45
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$\begingroup$ I show pictures on stats.stackexchange.com/a/324197/99274 higher moments also explained. $\endgroup$– CarlFeb 16, 2018 at 18:28
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If by graphical representation you meant histograms, I gather this is the best approach to provide a visual display of the moments. Please be aware we cannot specify any value for kurtosis. But there is a formula to provide kurtosis under random simulations.
Being k = kurtosis, sk = skewness, the formula is: k > sk^2 +1.
Below, a histogram with the specifications.
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1$\begingroup$ This is being automatically flagged as low quality, probably because it is so short. At present it is more of a comment than an answer by our standards. Can you expand on it? We can also turn it into a comment. $\endgroup$– Sycorax ♦Apr 12, 2021 at 0:53