# Central Tendency of Cauchy distribution

How do you measure the central tendency of a cauchy distribution?

I'm aware that the mean is not a good measure of the central location for cauchy. Can I use median?

I've been searching with the key word: central "tendency" of "cauchy" distribution stackexchange for a long time, and found no satisfactory answer that states how I can measure central tendency of Cauchy distribution. Everybody says that "there exists techniques that deals with...bla...bla..." that never gives a solid example of what there are.

Can someone give me an actual example?

• Wikipedia on Cauchy Dist'n under Estimation: "One simple method is to take the median value of the sample as an estimator of $x_{0}$ [center]... Other, more precise and robust methods have been developed. For example, the truncated mean of the middle 24% of the sample order statistics produces an estimate for $x_{0}$ that is more efficient than using either the sample median or the full sample mean. Commented Sep 23, 2019 at 6:42
• Also, because the density function is positive at the median, the distribution of the sample median is asymptotically normal for increasing sample size. Commented Sep 23, 2019 at 7:34
• My first search of this site for Cauchy tendency yielded several good answers. Variants of this search produce even more.
– whuber
Commented Sep 23, 2019 at 15:50

This may not be the best example of the claim in my Comment, but here are estimates of the center $$10$$ of a Cauchy with scale parameter 1.

Medians and trimmed means were found (in R) for a million samples of size $$n = 100.$$ The standard deviation of the trimmed means is a little smaller.

set.seed(922)
h = replicate(10^6, median(rcauchy(100,10)))
mean(h); sd(h)
[1] 9.999961
[1] 0.1585856
a.t = replicate(10^6, mean(rcauchy(100,10), tr=.38))
mean(a.t); sd(a.t)
[1] 10.00004
[1] 0.1535004


• Sampling doesn't seem to have anything to do with a question that is solely about a distribution.
– whuber
Commented Sep 23, 2019 at 14:52
• Pondered that, but "Everybody says that 'there exists techniques that deals with...bla...bla...' that never gives a solid example of what there are." made me think OP was asking about estimation. // For the dist'n, isn't the center just the median without further discussion? Commented Sep 23, 2019 at 15:46
• I, too, found the question to be unclear.
– whuber
Commented Sep 23, 2019 at 15:47