2.2A I NTRODUCTION TO N ORMAL D ISTRIBUTIONS. S ECTION 2.2A N ORMAL D ISTRIBUTIONS After this lesson, you should be able to… DESCRIBE and APPLY the 68-95-99.7.

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Presentation transcript:

2.2A I NTRODUCTION TO N ORMAL D ISTRIBUTIONS

S ECTION 2.2A N ORMAL D ISTRIBUTIONS After this lesson, you should be able to… DESCRIBE and APPLY the 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.

Normal distributions are good descriptions for some distributions of real data.

Normal distributions are good approximations of the results of many kinds of chance outcomes.

Many statistical inference procedures are based on Normal distributions.

WARNING: There is NO such animal as a normal curve in real life. Things are only approximately normal.

WARNING 2 : 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.

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.

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

Definition: The 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.

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.

The compression strength of concrete is approx normally distributed with mean 3000 psi and stdev 500.

The speed of a car involved in a fatal car crash is approx normally distributed with mean 51 mph and stdev 18 mph.

The heights of young American women are approx normally distributed with mean 64.5 inches and stdev 2.5 inches.