Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be normally distributed with mean $y/\log x$ and standard deviation $\sqrt{y(\log x/y)/(\log x)^2}$. But numerical testing produces a slight and systematic deviation from this conjecture. Specifically, I have tested for different interval lengths $y=x^c$. For each value of $c$, I calculated $\pi(x+y)-\pi(x)$, the number of primes in $[x,x+y]$, for $N$ non-overlapping intervals. Then I normalized the data by subtracting the mean and dividing by the standard deviation corresponding to each interval, as stated by the conjecture. We should therefore expect the resulting $N$ samples to be normally distributed with mean 0 and standard deviation $\sigma=1$. However, what I get is the following:

\begin{matrix} c& \sigma& N\\ \hline 0.20 & 0.967 & 240000\\ 0.25 & 0.966 & 240000\\ 0.30 & 0.965 & 240000\\ 0.40 & 0.958 & 240000\\ 0.50 & 0.947 & 240000\\ 0.55 & 0.917 & 40000\\ 0.60 & 0.899 & 20000\\ 0.65 & 0.891 & 10000\\ 0.70 & 0.889 & 5000\\ \end{matrix}

While each data set indeed are normally distributed, what appears to be happening is that the standard deviation decreases with increasing $c$, as compared to the conjecture by Montgomery and Soundararajan. The numerical results seem to be reasonably consistent, so I don't think they are an artifact of sampling only a finite number of intervals (however, please point out if I did any apparent mistakes in the above). I am not fluent enough in the theory to walk through Montgomery and Soundararajan's arguments myself, so I would greatly appreciate any comments on or explanation of this finding.

**EDIT** Including the lower order term in Montgomery and Soundararajan's conjecture, as suggested in the answer below by Lucia, we have the following revised numerics:

\begin{matrix} c& \sigma& N\\ \hline 0.20 & 1.000 & 240000\\ 0.25 & 1.001 & 240000\\ 0.30 & 1.003 & 240000\\ 0.40 & 1.002 & 240000\\ 0.50 & 1.000 & 240000\\ 0.55 & 1.001 & 40000\\ 0.60 & 0.992 & 20000\\ 0.65 & 0.995 & 10000\\ 0.70 & 1.009 & 5000\\ \end{matrix}

These numbers strongly support Montgomery and Soundararajan's conjecture, so it is clear that the missing lower order term was indeed responsible for the observed discrepancy.