1
Module 7 Percent Area and the Normal Curve
• What it is• History• Uses
2
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
3
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
4
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!
5
History of Normal Curve
• Fred Gauss (who cares about)– Laplace and DeMoive?
• Always looking up• Noticed that orbit• -estimates of planets– Were normally distributed
6
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
7
Use of the Normal Curve
• The normal curve always has the following proportions
8
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
9
Remember these Percentages …you will use them
• The normal curve always has the following proportions
10
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
11
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 %
12
Uses: % of Normal Curve
• What % have IQ between 85 and 115?- Between + and – 1 SD?- 34.13 + 34.13 = 68.26%