Sampling Distribution of Mean

Suppose $\mu$ is the population mean and $n$ samples are drawn, each having the mean of $\overline{x}_i$. The values of $x_i$ are the sampling distribution of mean.

Used to estimate population mean $\mu$, and calculate confidence intervals.

Mean

Average of all sample means $\overline{x}_i$, denoted as $\overline{x}$, is $\mu$.

Standard Error

Aka. standard deviation (not common though).

$$\sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}$$

Decreases as sample size increases.

Approx. to Normal Distribution

If the below conditions are met:

• The population is normally distributed OR the sample size is large enough.
• The population standard deviation $\sigma$ is known.

Then:

$$\overline{x} \sim N \left(\mu, \frac{\sigma} {\sqrt{n}} \right)$$

Note

Usually population’s standard deviation $\sigma$ is unknown. In such cases, Student’s t-distribution is used.