Why Student's t-distribution has fatter tails than normal distribution? Student's t-distribution that arise:

*

*when estimating the mean of a normally-distributed population

*when the sample size is small

*and the population's standard deviation is unknown

I need to clarify the Hypothesis H and Question Q:
H: In general you can control the tails of normal distribution with standard deviation, the bigger it is the fatter the tails so I can make the tails as fat as I like.
Q: Is this comparison based on the normal distribution used to produce the t-distribution?
It looks to me the answer is yes, for the previous question but maybe I am missing something important.

In general, and I understood this for the first time thanks to @whuber "a fat-tailed distribution" is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution according to Wikipedia
So my hypothesis H looks like is wrong at the very start, because you cannot alter the Normal distribution "fat tail" state with the standard deviation parameter.
It is pretty unclear why both the normal distribution or an exponential distribution are considered in here because their third and forth moment formula differs. In case of Exponentiation distribution 3rd and 4th moments are: 2 and 6, and in case of Normal distribution those moments are zero. So I wonder why Wikipedia has this definition of being "fat".


In my original idea of being on tail I would consider the area [-inf, $\mu -3\sigma]$ and $[\mu +3\sigma, \inf)$ and if some distribution has "fatter tail" would be to have large area under the curve in that tail area.

 A: Your "title" question seems more clear to me than your detailed question. So I ll try to answer the former.
Why is t-distribution fatter? because there is more uncertainty. Since we do not know the standard errors, there is more uncertainty of the t-statistic. However, the larger the sample is, the more accurate the Standard deviation estimate will be, thus t-distribution will converge with the normal distribution.
Now, you are partially correct. In a normal distribution, the larger the variance, the farther the tails will spread. However, a normal distribution has a very nice "ratio" of dispersion around the tails, and concentration around the means. In other words, once you standardized a variable, (so that it has a mean zero and standard deviation 1), if that variable follows a normal distribution, it will always look the same.
The t distribution "ratio" is larger than the one for normal distribution. So even if you standardize the variable, the distribution will depend on the degrees of freedom of the distribution. lower degrees of freedom, more uncertainty, fatter tails. Higher degrees of freedom, less uncertainty, t-distribution will look more and more like a normal distribution.
HTH
F
