Assume that accordning to a national statistic people in germany have a $mean$ of $10,000$ € on their bank account and we call that their "assets". Unfortunattely, only the $mean$ is given but not the standard deviation ($sd$). Assuming that the assets are normally distributed, is it possible to say what the $sd$ is? Note that in this example negative values are possible and would mean that someone has debts (maybe there is a more correct word than assets for what I mean but I don't know).
My thoughts so far:
A standard normal distribution has a mean of $0$ and a sd of $1$. Now we can think of the distribution of the assets as a standard normal distribution that has been shifted by $+10,000$. The resulting distribution has $mean$ of $10,000$ and a $sd$ of $1$ and is normally distributed. The problem with that idea is that every distribution can be seen as a shifted normal distribution, hence we would always assume a $sd$ of $1$. This neglects the scale of the variable. For example, for a $mean$ of $10,000$ € the $sd$ would $1$ and for a $mean$ of $1,000,000$ cents the $sd$ would be $1$, too. This doesn't seem correct to me.
Because of the answer I want to further explain why I was thinking it would be possible to tell the sd. We call only distributions with certain shapes a normal distribution, i.e. the distribution must have a kurtosis of 3 and a skewness of 0 (both approximately). The sd reflects the shape of a distribution, too, hence, I thought a distribution can't be normal with any sd (or does it?). This lead me to the idea that there must be some sd that the distribution must have if it is supposed to be a normal distribution. What is the mistake in my reasoning?
I changed the example because the variable of the first example can not be normal, hence, the question wouldn't make that much sense (as pointed out by Glen_b).