Similar to another thread I've posted, only that one's asking how to sketch the functions, this time I'm stumped on the calculations.

So the full Question (apart from part (i) because I've done that, and it doesn't relate) is...

*A manufacturer of watches claims that the weekly error, in seconds, of their watches follow a $\displaystyle N(0,1)$ distribution. An inspection of the performance of 5 randomly chosen watches gave the following weekly errors $\displaystyle -0.68$, $\displaystyle 0.25$, $\displaystyle 0.29$, $\displaystyle -1.41$ and $\displaystyle 1.59$.*

**(ii)** Assume that the weekly errors follow a $\displaystyle N(0,\sigma^{2})$ distribution. Sketch the likelihood function and log-likelihood for $\displaystyle \sigma$ and calculate the most likely value of $\displaystyle \sigma$.

All I've got so far is

$\displaystyle f_{X}(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\{{-\frac{1}{2}(\frac{x}{\sigma})^{2}}\}$

But then anything could be wrong after that

I got this for the likelihood function:

$\displaystyle \sigma^{-n}(2\pi)^{-n/2}\exp\{-\frac{1}{2}\frac{\sum_{i=1}^{n}x_{i}^{2}}{\sigma^{ 2}}\}$

Which gave me the log-likelihood function of:

$\displaystyle K-n\ln{\sigma}-\frac{\sum_{i=1}^{n}x_{i}^{2}}{2\sigma^{2}}$

And now I'm stumped. I don't think what I've done is right at all. Can anyone help?