4.4: Standard Normal Distribution

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Learning Objectives

• State the mean and standard deviation of the standard normal distribution
• Use a $$Z$$ table
• Use the normal calculator
• Transform raw data to $$Z$$ scores

As discussed in the introductory section, normal distributions do not necessarily have the same means and standard deviations. A normal distribution with a mean of $$0$$ and a standard deviation of $$1$$ is called a standard normal distribution.

Areas of the normal distribution are often represented by tables of the standard normal distribution. A portion of a table of the standard normal distribution is shown in Table $$\PageIndex{1}$$.

Table $$\PageIndex{1}$$: A portion of a table of the standard normal distribution
Z Area below
-2.5 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 0.008
-2.4 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 distribution; the second column contains the area below $$Z$$. 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) the mean. For example, a $$Z$$ of $$-2.5$$ represents a value $$2.5$$ standard deviations below the mean. The area below $$Z$$ is $$0.0062$$.

The same information can be obtained using the following Java applet. Figure $$\PageIndex{1}$$ shows how it can be used to compute the area below a value of $$-2.5$$ on the standard normal distribution. Note that the mean is set to $$0$$ and the standard deviation is set to $$1$$.

Calculate Areas

A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following 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.

Example $$\PageIndex{1}$$

As a simple application, what portion of a normal distribution with a mean of $$50$$ and a standard deviation of $$10$$ is below $$26$$?

Solution

Applying the formula, we obtain

$Z = \frac{26 - 50}{10} = -2.4$

From Table $$\PageIndex{1}$$, we can see that $$0.0082$$ of the distribution is below $$-2.4$$. There is no need to transform to $$Z$$ if you use the applet as shown in Figure $$\PageIndex{2}$$.

If all the values in a distribution are transformed to $$Z$$ scores, then the distribution will have a mean of $$0$$ and a standard deviation of $$1$$. This process of transforming a distribution to one with a mean of $$0$$ and a standard deviation of $$1$$ is called standardizing the distribution.

This page titled 4.4: Standard Normal Distribution is shared under a Public Domain license and was authored, remixed, and/or curated by David Lane via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.