TITLE: Why does the Cauchy distribution have no mean if it's symmetric around 0?
QUESTION [12 upvotes]: Something that didn't make intuitive sense to me when learning about the Cauchy distribution was that there was no defined mean for the function, even though the function was clearly centered at zero and equally valued in both directions.
Is there any reason for this?

REPLY [6 votes]: Just to add a clear demonstration of what actually happens when you try to deal with the mean and higher moments of a Cauchy distribution, I ran a quick script to repeatedly take n samples of a standard Cauchy distribution (5 runs for each n):
n = 1000
    mean = -1.02224, sd = 22.0379
    mean = 0.443686, sd = 18.5603
    mean = -0.616193, sd = 20.8578
    mean = 0.544703, sd = 16.2545
    mean = 1.99947, sd = 56.7486
n = 10000
    mean = 0.20199, sd = 41.3423
    mean = 3.47629, sd = 364.8
    mean = -1.4106, sd = 80.6524
    mean = -0.441166, sd = 224.783
    mean = -0.674296, sd = 66.4877
n = 100000
    mean = 1.13362, sd = 413.799
    mean = -1.06265, sd = 228.098
    mean = 1.09204, sd = 317.432
    mean = 3.80845, sd = 1493.95
    mean = -0.377224, sd = 295.982
n = 1000000
    mean = -1.41118, sd = 3189.89
    mean = -1.66183, sd = 1797.63
    mean = -0.176471, sd = 422.138
    mean = 1.30805, sd = 2023.47
    mean = 0.723504, sd = 1575.73

You can plainly see how the mean and SD jump crazily all over the place, and show no sign of converging on any kind of meaningful value.