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This module introduces standard normal distribution and standardizing the 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] .

A portion of a table of the standard normal distribution.
Z Area left of Z
-2.50 0.0062
-2.49 0.0064
-2.48 0.0066
-2.47 0.0068
-2.46 0.0069
-2.45 0.0071
-2.44 0.0073
-2.43 0.0075
-2.42 0.0078
-2.41 0.0080
-2.40 0.0082
-2.39 0.0084
-2.38 0.0087
-2.37 0.0089
-2.36 0.0091
-2.35 0.0094
-2.34 0.0096
-2.33 0.0099
-2.32 0.0102

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 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.

An example from the applet.
Calculate Areas

A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using thefollowing formula: z x μ σ where z is the value on the standard normal distribution, x is the value on the original distribution, μ is the mean of the original distribution and σ 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 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] .

Area below 36 in a normal distribution with a mean of 50 and a standard deviation of 10.
If all the values in a distribution are transformed to z -scores, then the distribution will have a mean of 0 and a standard distribution. This process oftransforming a distribution to one with a mean of 0 and a standard deviation of 1 is called standardizing the distribution.

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Source:  OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5
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