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Chapter 3: The Normal Distributions

Knowledge Prerequisites

Graphing and describing distributions

1. declarative knowledge (definitions)
   1. distribution
   2. histogram
   3. shape, center and spread of a distribution
   4. skewness of a dsitribution
   5. mean, $\overline{x}$
   6. median, M
   7. standard deviation, $s$
2. procedural knowledge
   1. how to construct a histogram
3. conditional knowledge
   1. know which measures of center and spread are appropriate for any given set of data
4. resources
   1. Chapter 1 of textbook
   2. Chapter 2 of textbook

Learning Goals

1. declarative knowledge (definitions)
   1. properties of density curve
   2. normal density curve
   3. normal distribution, $N(\mu ,\sigma )$
   4. standard normal distribution, $N(0,1)$
   5. population mean, $\mu$
   6. population standard deviation, $\sigma$
   7. 68-95-99.7 rule
   8. $z$-score
   9. resistant measure
   10. 1.5 IQR rule (interquartile range)
2. procedural knowledge
   1. how to find standard deviation from the graph of a normal curve
   2. how to use the 68-95-99.7 rule to find the proportion (i.e., percentage) of area under a normal curve without a calculator
   3. how to find the proportion (i.e., percentage) of area under a normal curve using NDAREA program on TI-83
   4. how to find a value given a proportion of area under a normal curve
   5. how to find variance of a set of data
   6. how to find the $z$-score
3. conditional knowledge
   1. know the difference between mean and median and what they say about a distribution.
   2. know where to find mean and median on a density curve.
   3. know the purpose of the standard normal curve
   4. know the difference between population mean ($\mu$) and sample mean ($\overline{x}$)
   5. know the difference between population standard deviation ($\sigma$) and sample standard deviation ($s$)

Questions to Ponder

1. What is the best way to study/prepare for the concepts in this section? Are there are other methods to study these concepts?
2. What questions do you need to ask the instructor?

Purpose of this Section

1. To examine a particular distribution (i.e., the normal distribution). The normal distribution is a very important distribution that will be used many times in this course.

General Notes

Density Curves

• Density curves are mathematical models for a distribution. They provide an overall pattern and ignore any irregularities and outliers. Another reason that density curves are preferred over histograms is that histograms are affected by class width.
• As we make the class size smaller and increase the number of observations, the histogram starts to look like the smooth density curve.
• No set of real data is exactly described by a density curve. Even so, it is an approximation that provides us useful details about the distribution.
• Because the density curve is an idealized description of a distribution, we distinguish the mean (now represented by $\mu$) and the standard deviation (now represented by $\sigma$) in the density curve from the mean and standard deviation ($\overline{x}$ and $s$) of the distribution.

Normal Density Curves

You can explore how the mean and standard deviation affect the normal curve.

Use the TI-83 program called NRMHST to check if a set of data is normally distributed.

68-95-99.7 Rule

In the normal distribution with mean, $\mu$, and standard deviation, $\sigma$ (i.e., $N(\mu ,\sigma )$):

• approximately 68% of the observations fall between $\mu -\sigma$ and $\mu +\sigma$;
• approximately 95% of the observations fall between $\mu -2\sigma$ and $\mu +2\sigma$;
• approximately 99.7% of the observations fall between $\mu -3\sigma$ and $\mu +3\sigma$;
• you can explore the 68-95-99.7 rule.

Standard Normal Distribution, $N(0,1)$

Since all normal distributions have the same properties, we can standardize the mean and standard deviation.

A $z$-score is a standardized observation. Let $x$ be an observation from a distribution that has mean, , and standard deviation, , then the standardized observation is given by:

A $z$-score tells us how many standard deviations the original observation falls from the mean and in which direction.

Normal distribution calculations

NOTE: you are NOT expected to use Table A of the textbook to answer any questions in the text or on a test. You are required to become familiar with how to use the TI-83 to calculate the answer.

An area under the curve provides the proportion (in percent or decimal form) of the observations in a distribution. Since all normal distributions are the same after standardization, we can find the area under any normal curve from a single table (Table A on pp. 652-3). You MUST standardize before using the table. Also, the table only provides areas to the left of a given $z$-score. The area between a and b is equal to the proportion of the observations between $a$ and $b$. To calculate the area by hand, follow examples 3.6 and 3.7 on pp. 69-70.

However, the TI-83 provides an easy function to determine the areas!!!! Here are the steps. You must remember that the number that the calculator gives you is the area under the normal curve which represents a proportion (decimal form) of the observations satisfying the given conditions.

Even though the calculator provides the proportion, you may be asked to sketch the area. Use the TI-83 program called NDAREA to to see a graph of the normal distribution.

Finding a value given a proportion

We may want to find the observation above which (or below which) a given proportion exists. In other words, given an area, what is the observation that has the given area to the right (or left). This is slightly more difficult to do using Table A. Luckily, the TI-83 calculator provides a built-in function to find the value of the observation!!!! Here are the steps.

Resources

• Working with $z$-scores and Normal Probabilities - http://psych.colorado.edu/~mcclella/java/normal/normz.html - This applet converts between raw scores and $z$-scores with a display of various areas of probability.
• The Normal Distribution - http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html - This applet draws a normal curve with a specified mean and standard deviation and demonstrates the 68-95-99.7 rule for normal distributions.
• Standard Normal Distribution - http://www.coe.tamu.edu/~strader/Mathematics/Statistics/NormalCurve/ - This applet is designed to enhance your understanding of standard scores, areas under the normal curve, and probability as applied to inferential statisitics.
• Probabilities for the Normal Distribution - http://psych.colorado.edu/~mcclella/java/normal/handleNormal.html - This applet applet may be used to find approximate probabilities from the normal distribution.

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