Measuring tails of a normal distribution - kurtosis and variance

Question

So it seems that kurtosis is the way to measure how heavy the tails are in a normal distribution. But I was wondering if variance could also be a way of measuring how heavy the tails are. So I'm trying to come up with a normal distribution with high variance but small tails and I can't think of one. Because if the mass varies highly from the mean, it must be in the tails, right?

Answer 1

Could you clarify a bit what you're looking for? If the distribution is normal, then you are reasonably limited in what you can do with the shape of the curve.

You might consider the 5th moment, which is a measure of tail asymmetry--2nd moment is variance, 3rd is skew, and 4th is kurtosis. I played around with generating distributions in R for the statistics class I teach, to use as examples, and found the code here to be incredibly helpful:
How to generate distributions given, mean, SD, skew and kurtosis in R?):

The following link has some useful thoughts.
https://stats.stackexchange.com/questions/2893/moments-of-a-distribution-any-use-for-partial-or-higher-moments

Answer 2

Kurtosis is not a measure for heavy tails in normal dist since all normal dists no matter what the variance is have zero kurtosis. Actually kurtosis measures how close is your distribution to a normal one.