Can any data be transformed into a standard normal distribution? There is a standardized transformation which can transform a normal distribution to standard normal distribution:
$$x^{(i)}=\frac{x^{(i)}-\mu_x}{\sigma_x}$$
I am wondering given a uniform distribution or any other distribution, can we transform it into a standard normal distribution using the above equation?
It is difficult to see from the following codes.
import numpy as np
import matplotlib.pyplot as plt

x = np.random.randint(0, 10000, (1, 100))
y = np.random.randint(0, 4000, (1, 100))

z = np.random.randn(1, 100)

x_s = (x - x.mean()) / x.std()
y_s = (y - y.mean()) / y.std()

# plt.hist(x_s, bins=1)
# plt.show()

import seaborn as sns

# sns.distplot(x, rug=True, bins=None)
sns.distplot(x_s, rug=True, bins=None)
# sns.distplot(z, rug=True, bins=None)

plt.show()

 A: Adding or subtracting a value from a random variable (or set of samples) will shift the distribution. Multiplying or dividing by a value will scale it. Standardizing (i.e. subtracting the mean and dividing by the standard deviation) will force the mean to zero and the standard deviation to one. But, this is just a particular way of shifting and scaling; it can't change the fundamental shape of the distribution. So, if the distribution isn't normal to begin with, standardizing won't make it normal.
For example, here's an example of a non-normal distribution before and after standardizing. You can see that it has been shifted and scaled, but the shape is the same:
A: Let suppose you want to find the density $f_{z}(z_i)$ of the random variable 
$$
z_i = \frac{x_i-\mu_x}{\sigma_x}
$$
where $\mu_x = E(x_i)$ and $\sigma_x^2 = Var(x_i)$.
Using the rule of variable transformation we have that 
$$
f_{z}(z_i) = f_{x}(x_i(z_i))|\frac{\partial x_i(z_i)}{\partial z_i}|
$$
where $f_{x}()$ is the density of $x_i$,  and $x_i(z_i)$ is $x_i$ written as a function of $z_i$, i.e.
$$
x_i = z_i\sigma_x+\mu_x
$$
We have also that
$$
|\frac{\partial x_i(z_i)}{\partial z_i}| = \sigma_x.
$$
It is easy to see that this produce a standard normal only if $f_x()$ is normal.
