The sampling distribution, i.e. the distribution of the means of the individual samples is NOT the normal distribution, as this blog states. The sampling distribution is the so-called Student t distribution, which, although similar in shape to the normal (Gaussian) (bell-shaped) has different parameters. In particular, normal distribution does not depend on the sample size while the Student t distribution does (sample size / degrees of freedom is a parameter of Student t). The spread of normal is characterized by variance (standard deviation squared) while the spread of Student t is characterized by the standard error squared.