# What is to median as first central moment is to mean?

The question sounds like a riddle, but it isn't intended to be one.

I've been thinking about the Cauchy Distribution which, famously doesn't have any central moments defined. A very informal justification for this is that as the angle approaches $\pm90^\circ$ from the origin, the value of the function tends quickly to infinity... hence, if we were to attempt to calculate the mean, its value would vary to $\pm\infty$ very, easily. Essentially, rather than summarise the data-set as a whole, one would identify only whether or not your samples were biased very slightly to the positive or negative values.

An obvious approach to establish an estimate of expected value would be to calculate the median - which would avoid the outlying data points overwhelming the summary. This single scalar summary value - analogous to mean - then suggests a more reasonable estimate of 'expected' value in some circumstances. Is it common to extend such analysis with measures analogous to variance, skew and kurtosis - to better describe the distribution? If so, how are these concepts commonly defined?

UPDATE: Many thanks for the pointer to MAD... that's definitely relevant. While I wasn't clear about this previously, central moments appealed because they generated a progression of values each further refining the description of a normal distribution... and I really hoped to do something similar for systems where the empirical mean and standard deviation can't be trusted to give a meaningful summary.

• Supposing one tries to find the mean: $$\operatorname E X = \int_{-\infty}^\infty x \cdot \frac{dx}{1+x^2}.$$ One has \begin{align} & \lim_{a\to+\infty} \int_0^a x\cdot\frac{dx}{1+x^2} = +\infty \quad\text{and} \quad \lim_{a\to+\infty} \int_{-a}^0 x \cdot \frac{dx}{1+x^2} = -\infty \\ \\ \text{and } & \lim_{a\to+\infty} \int_{-a}^a x\cdot\frac{dx}{1+x^2} =0 \ne \log_e 2 = \lim_{a\to+\infty} \int_{-a}^{2a} x\cdot\frac{dx}{1+x^2} \end{align} and also the strong law of large numbers fails to hold, and in contrast to the central limit theorem situation, the distribution of$\,\ldots\qquad$ Jul 21, 2018 at 18:54
• $\ldots\,$of the mean of an i.i.d. sample has just as much dispersion as the distribution of each obervation in the sample -- it doesn't get narrower as the sample size grows. In fact the mean of the sample has the same distribution as the mean of each observation in the sample. $\qquad$ Jul 21, 2018 at 18:55
• Thus the proposed "very informal justification" is far too weak an indictment. Jul 21, 2018 at 18:56

There are various measures of variability that fit your description. One that is popular in some fields is the inter-quartile range.

Another one is median absolute deviation. Charmingly, the standard acronym for this is MAD.

• ...though sadly the standard acronym for mean absolute deviation is also MAD. Jan 17, 2012 at 0:37

If the median is used as an indication of central tendency, the interquartile range (i.e the difference between third and first quartiles) and the median absolute deviation can be used as indications of dispersion. See http://en.wikipedia.org/wiki/Robust_measures_of_scale

• Thanks especially for the Wiki link. That nails what I was looking for perfectly. Jan 17, 2012 at 0:16
• Side comment: there are several variants of the MAD out there. Examples: Wholly mad definition: "Median of all absolute deviations around the median'' versus mildly mad: "Mean of all absolute deviations around the mean." I find the latter very useful when introducing the variance to students. No doubt Robert Israel was thinking of the "wholly MAD" definition (median of median). For completeness, here are two half-mad definitions: "Median of all absolute deviations around the mean" and "Mean of all absolute deviations around the median." Feb 14, 2018 at 8:25
• I just noticed Henry's comment below, which is very similar to my comment. Feb 14, 2018 at 8:27

The first moment is the mean. The first central moment is always $0$, when it exists.

The Cauchy distribution has neither a mean nor a standard deviation, but it does have location and scale parameters: the median and the interquartile range.

One consequence of the unusual (if you used to thinking only about distributions with moments) behavior of this distribution is that the sample mean $(X_1+\cdots+X_n)/n$ has the same distribution, with just as large a scale parameter as a single observation $X_1$. Maybe the easiest way to prove that is by looking at the characteristic functions $t\mapsto E(e^{itX_1})$ and $t\mapsto E(e^{it(X_1+\cdots+X_n)/n})$.