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Module 7 Percent Area and the Normal Curve

• What it is• History• Uses

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Normal Curve Characteristics

• Inflection points (at + and – 1 SD)– Where slopes changes from down to out.

• Axes – X –axis (abscissa) =Scores (as usual)– Y –axis (ordinate) = freq of scores or %

• Asymptotic– Tails never touch abscissa– Allows for extreme scores

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The Normal Curve

• The normal curve is symmetric, bell shaped, and asymptotic

• The inflection points fall at one standard deviation above and below the mean

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Normal Curve

• Theoretical distribution– If an infinite number of observations were collected

• But smaller Ns distribute themselves normally– But only IF….the underlying population is normally distributed!

• Ns of 30 to 40 are usually enough• N of a few hundred is plenty!

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History of Normal Curve

• Fred Gauss (who cares about)– Laplace and DeMoive?

• Always looking up• Noticed that orbit• -estimates of planets– Were normally distributed

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Sir Francis Galton

• Noticed that IQ is normally distributed– In the population

• And so is practically everything else – Psychological – Physical (height, weight)– Behavioral (achievement, sexual behavior) – Gun shots at a target (or person!)– As long as the events are independent

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Use of the Normal Curve

• The normal curve always has the following proportions

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Uses

• But real work events don’t always play by the rules – Because many are not independent– Can you think of some examples• (Think about things that are related)

• Nevertheless …the Normal Curve is still useful– For real world “lumpy” or skewed distributions – i.e. “robust” to minor violations of shape

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Remember these Percentages …you will use them

• The normal curve always has the following proportions

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Uses con’t

• Look at p 92 figure 7.4• What are the Ms an SDs for:– IQ score?• M = 100; SD =15

– SAT score?• M =500; SD = 100

– Height (US adult males)• M = 69.5 in; SD = 2 inches

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Uses con’t

• With the known M and SD – We can use the percentages(under the curve)• To interpret INDIVIDUAL scores • E.g. the relative number of those scoring in porportoins

of the curve

– What % of males are taller than 6’ 3 ½”? (75.5 in)• 0.13% (just a very few)…less tha 1/10 percent• Notice that includes everyone below that height

– Taller than 99.47 %

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Uses: % of Normal Curve

• What % have IQ between 85 and 115?- Between + and – 1 SD?- 34.13 + 34.13 = 68.26%