2.2A Introduction to Normal Distributions

Section 2.2A Normal Distributions After this lesson, you should be able to… DESCRIBE and APPLY the 68-95-99.7 Rule DESCRIBE the standard Normal Distribution

One particularly important class of density curves are the Normal curves, which describe Normal distributions. All Normal curves are symmetric, single-peaked, and bell-shaped A specific Normal curve is described by giving its mean µ and standard deviation σ.

Two Normal curves, showing the mean µ and standard deviation σ. Have students practice sketching different versions of the curve. Then find the inflection point to estimate the standard deviation.

Normal distributions are good descriptions for some distributions of real data. Some examples of real data: the distribution of all the heights of adult women, the circumference of ball bearings produced in a factory, etc.

Normal distributions are good approximations of the results of many kinds of chance outcomes. Some examples of chance outcomes: Flipping a coin 200 times and counting the number of times that you get heads---running 1000’s of trials and looking at the distribution of the number of heads in 200 flips, etc.

Many statistical inference procedures are based on Normal distributions. Almost all the hypotheses tests we will run next semester…

There is NO such animal as a normal curve in real life. WARNING: There is NO such animal as a normal curve in real life. Things are only approximately normal. The formula for the normal curve is coming up….

Because that’s what they are. WARNING2: You must use the adjective approximately to describe normal distributions. Because that’s what they are. Don’t make me use my red pen on your work. Emphasize to students that they must claim APPROX normality always.

Just to show the mathy people…

I would like you to use approx as well. A Normal distribution is described by a Normal density curve. Any particular Normal distribution is completely specified by two numbers: its mean µ and standard deviation σ. The mean of a Normal distribution is the center of the symmetric Normal curve. The standard deviation is the distance from the center to the change-of-curvature points on either side. (Hello Calculus Fairy) We abbreviate a Normal distribution with mean µ and standard deviation σ as N(µ,σ). I would like you to use approx as well. The change of curvature point is the inflection point we physically found in slide 12.

Two Normal curves, showing the mean µ and standard deviation σ.

Definition: The 68-95-99.7 Rule (“The Empirical Rule”) In a Normal distribution with mean µ & standard deviation σ: Approximately 68% of the observations fall within σ of µ. Approximately 95% of the observations fall within 2σ of µ. Approximately 99.7% of the observations fall within 3σ of µ.

Glue In

After measuring many, many mothers, the authors of a biology paper published in 1903 argued that the distribution of their heights was approx normally distributed with a mean of 62.5 and a stdev of 2.4 inches. Have the students draw an approx normal curve and mark the three stdev on each side then ask about 68%, etc. What percent are greater than 69.7 inches? What percent are greater than 60.1 inches, etc.

So…might it be possible to use a z-score to claim outlier status?

The compression strength of concrete is approx normally distributed with mean 3000 psi and stdev 500. Ditto the same as you did with slide 15

The speed of a car involved in a fatal car crash is approx normally distributed with mean 51 mph and stdev 18 mph. Am I telling the truth? (No, because you cannot get to 3 stdev below the mean without having a negative speed.) Watch out for this.

The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches. http://digitalfirst.bfwpub.com/stats_applet/generic_stats_applet_7_norm.html