Does a uniform random distribution become a normal distribution? I have $N$ numbers from uniform random distribution. Now I have done a transformation on these $N$ numbers as: each $x$ is converted to $\frac{(x-\mu_N)}{\sigma_N}$. Where $\mu_N$ is the mean of $N$ numbers and $\sigma_N$ is the standard deviation of those $N$ numbers. Does the converted distribution follow a normal distribution?

In order for your sample to follow a normal distribution, you may want to feed it to a normal distribution's quantile function, which is:
$\mu+\sigma\sqrt2 erf^{-1}(2x-1)$ where $erf^{-1}$ is the inverse error function. Or you could simply feed you sample into the function qnorm() in R:
N <- 1000
sample <- runif(1000)
normal <- qnorm(sample, mean=10, sd=4)
hist(normal)

In both cases, you choose the mean ($\mu$) and standard distribution ($\sigma$) yourself.

