5.3 The standard normal distribution

As previously discussed, normal distributions do not necessarily have the same means and standarddeviations. A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution    and is typically represented by $Z$ .

Areas underneath the normal distribution are often represented by tables of the standard normal distribution. A portion of a table of thestandard normal distribution is shown in [link] .

The first column titled $Z$ contains values of the standard normal random variable; the second column contains the area below the curve to the left of $z$ . Since the distribution has a mean of 0 and a standard deviation of 1, the $Z$ column is equal to the number of standard deviations below (or above) themean. For example, a $z$ of -2.5 represents a value 2.5 standard deviations below the mean. Thearea below the curve to the left of $\mathrm{z=-2.5}$ is 0.0062.

The same information can be obtained using a calculator or the following Java applet. [link] shows how it can be used to compute the area below the standard normal curve to the left of -2.5. Note that the mean is set to 0 andthe standard deviation is set to 1.

A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using thefollowing formula: $$z=\frac{x-\mu }{\sigma }$$ where $z$ is the value on the standard normal distribution, $x$ is the value on the original distribution, $\mu$ is the mean of the original distribution and $\sigma$ is the standard deviation of the original distribution.

As a simple application, what portion of a normal distribution with a mean of 50 and a standard deviation of 10 is below26? Applying the formula we obtain $$z=\frac{25-50}{10}=-2.4$$

From [link] , we can see that 0.0082 of the distribution is below -2.4. There is no needto transform to $Z$ if you are using a technology or the applet shown in [link] .