e-Statistics

Z-score

When a data set is assumed to be governed by a normal density function $ f(x)$ with parameters $ \mu =$ and $ \sigma =$ , we say ``it is normally distributed with mean $ \mu$ and standard deviation $ \sigma$,'' for which we often simply write $ N(\mu,\sigma^2)$. In particular we call $ N(0,1)$ the standard normal distribution. Suppose that a value $ X =$ in data is normally distributed with $ (\mu,\sigma)$ (here $ X$ is also referred to a ``normal random variable''). Then
$ Z = \displaystyle\frac{X - \mu}{\sigma} =$
is called the z-score, and the distribution of $ Z$ becomes the standard normal distribution $ N(0,1)$. Standard normal distribution table contains the area under the standard normal curve from $ Z=0$ to $ Z=z$. This table can be used to compute the probability involving a normal random variable.

Example. Suppose that you scored 650 on SAT in 2000, and 30 on ACT in 2001. The SAT exam in 2000 had mean 500 and standard deviation 100, and the ACT had mean 21 and standard deviation 4.3. How can you compare these scores? Can you say you did it better in 2001?