Understanding central limit theorem What is wrong with the following sentence: 

The Central Limit Theorem  implies that, as the sample size grows, the error distribution approaches normality.

Am I correct by saying that it should in stead state the MEAN of the sample error approaches zero as sample size grows?
 A: In its standard simplest form, the Central Limit Theorem (CLT) is a statement 
about the cumulative distribution function of the random variable
$$Z_n = \frac{X_1 + X_2 + \cdots + X_n -n\mu}{\sigma \sqrt{n}}$$
where the $X_i$ are independent identically distributed random variables
with mean $\mu$ and standard deviation $\sigma$.  The CLT asserts that
for each $a$, $-\infty < a < \infty$,
$$F_{Z_n}(a) = P\left\{\frac{X_1 + X_2 + \cdots + X_n -n\mu}{\sigma \sqrt{n}}
\leq a
\right\} \to \Phi(a) = \int_{-\infty}^a \frac{e^{-x^2/2}}{\sqrt{2\pi}}\mathrm dx$$
as $n \to \infty$.
If by "error distribution" you mean the distribution function of
$$Y_n = \left(\frac{1}{n}\sum_{i=1}^n X_i\right) -\mu
= \frac{\sigma}{\sqrt{n}}Z_n,$$
that is, the difference of the sample mean $\bar{X} = n^{-1}\sum_iX_i$ 
and the population mean $\mu$, then the CLT certainly
does not imply that $F_{Y_n}(\cdot)$ "approaches normality" as the
sample size $n$ grows large in the
usual sense of normality, though nitpickers may want to claim that the
distribution is approaching a normal distribution with mean $0$ and 
standard deviation $0$ (often called a constant by statistically 
illiterate people).
On the other hand, the mean of the sample error is not a random
variable but a constant (in fact, $0$ since the sample mean is an unbiased
estimator of the population mean) and does not need to approach
$0$; it is already there!  I think what you meant to say is that the
distribution $F_{Y_n}(a)$ of the sample error approaches the unit
step function:
$$F_{Y_n}(a) \to u(a) = \begin{cases}1, & \text{if}~a > 0,\\
0 &\text{if}~a < 0,\end{cases}$$
which is certainly correct, and follows from the CLT,
but also follows from results such as
the weak law of large numbers which makes no assertions
about the distribution of $Z_n$, only about $Y_n$.
