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Do sampling distribution and sampling from distribution mean the same thing? I am interested in x~N($\mu$, $\sigma$).

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Sampling from a distribution (verb) refers to the process of drawing a random value (or set of values) from a probability distribution. Your example $X \sim \mathcal{N}(\mu, \sigma)$ means that random variable $X$ is normally distributed (with the specified parameters). Generating a realization (i.e. particular value) of $X$ would be sampling from this normal distribution.

A sampling distribution (noun) is the probability distribution of a statistic computed from a random sample. That is, suppose we have a dataset containing $n$ points. The data were obtained by randomly sampling from some population or underlying probability distribution. The statistic is a quantity computed from the data (e.g. the mean). Imagine performing the same experiment (infinitely) many times: sample a new dataset and compute the statistic. The distribution of the statistic across these repetitions is the sampling distribution.

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