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Bookmark or Share- Why does the Cauchy distribution have no mean?What is very interesting is that the existence of a zero principal value might tempt us to define the mean of the Cauchy distribution (or the mean of any RV) as the principal value of the integral. This probes much more deeply into the nature of this question, which is glossed over by declaring that the integral is either infinite or undefined ...
- mean - Is Median Absolute Percentage Error useless ... - Cross ValidatedThe mean on the other hand, gives a more weighted information about the entire distribution and includes the tails. $$\text{mean}(X) = 1- \int_0^\infty F(x) dx$$ Let's look at a few curves with median or mean equal to 1. The red curves have an average of 1 and will be restricted to be below the black curve 1/x.
- Which "mean" to use and when? - Cross ValidatedSo we have arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM). Their mathematical formulation is also well known along with their associated stereotypical examples (e.g., Harmonic mean and it's application to 'speed' related problems).
- What does standard deviation tell us in non-normal distributionWith median then it makes a sense to talk about quantiles, which could be a way to go with skewed distributions. OP didn't focus on skewed distributions though. So, for any symmetrical distribution you have mean=median, it doesn't have top be normal. Thus it makes a sense to talk about mean when standard deviation is discussed. $\endgroup$ –
- spss - What is mean rank? - Cross ValidatedMean rank will be the arithmetic average of the positions in the list: $$\frac{1.5+1.5+3+4+5}{5}=3$$ When there is an odd number of rows, the median will be the middle value of the original data after it is ranked.
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