Confusion with Z-Score

Question

Having some issue with the concept of Z score.

When exactly do I use $Z = \frac{\bar X - u}{\sigma}$,

and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$.

I get very confused with in what situation should be using which Z calculation. Really appreciate it to have someone explain the concept. Cheers.

Answer 1

Suppose you have $X\sim N(\mu,\sigma^2)$ then you would use $Z=\frac{X-\mu}{\sigma}$.

The $Z$ score tells you how many standard deviations $X$ is from the mean.

If you took a sample of size $n$ then you might want to look at the distribution of the sample mean denoted $\overline{X}$. The central limit theorem says that $\overline{X}\sim N(\mu,\frac{\sigma^2}{n})$. This is known as the sampling distribution of means. The standard deviation of this distribution is $\frac{\sigma}{\sqrt{n}}$ and called the standard error. Hence standardizing $\overline{X}$ leads to $Z=\frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{n}}}$.

Hope this helps.

Answer 2

You use $z=\frac{\overline{x}-\mu_{\overline{x}}}{\sigma_{\overline{x}}}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ when you are dealing with a sampling distribution. That is, when your data points are means of samples of size $n$. You use $z=\frac{x-\mu}{\sigma}$ in a normal distribution to tell you how many standard deviations a given $x$ is from $\mu$.