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The Normal Distribution. History. Abraham de Moivre (1733) – consultant to gamblers Pierre Simon Laplace – mathematician, astronomer, philosopher, determinist. Carl Friedrich Gauss – mathematician and astronomer. Adolphe Quetelet -- mathematician, astronomer , “social physics.”.

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history
History
  • Abraham de Moivre (1733) – consultant to gamblers
  • Pierre Simon Laplace – mathematician, astronomer, philosopher, determinist.
  • Carl Friedrich Gauss – mathematician and astronomer.
  • AdolpheQuetelet -- mathematician, astronomer, “social physics.”
importance
Importance
  • Many variables are distributed approximately as the bell-shaped normal curve
  • The mathematics of the normal curve are well known and relatively simple.
  • Many statistical procedures assume that the scores came from a normally distributed population.
slide4

Distributions of sums and means approach normality as sample size increases.

  • Many other probability distributions are closely related to the normal curve.
using the normal curve
Using the Normal Curve
  • From its PDF (probability density function) we use integral calculus to find the probability that a randomly selected score would fall between value a and value b.
  • This is equivalent to finding what proportion of the total area under the curve falls between a and b.
the pdf
The PDF
  • F(Y) is the probability density, aka the height of the curve at value Y.
  • There are only two parameters, the mean and the variance.
  • Normal distributions differ from one another only with respect to their mean and variance.
avoiding the calculus
Avoiding the Calculus
  • Use the normal curve table in our text.
  • Use SPSS or another stats package.
  • Use an Internet resource.
iq 85 pr
IQ = 85, PR = ?
  • z = (85 - 100)/15 = -1.
  • What percentage of scores in a normal distribution are less than minus 1?
  • Half of the scores are less than 0, so you know right off that the answer is less than 50%.
  • Go to the normal curve table.
normal curve table
Normal Curve Table
  • For each z score, there are three values
    • Proportion from score to mean
    • Proportion from score to closer tail
    • Proportion from score to more distant tail
locate the z in the table
Locate the |z| in the Table
  • 34.13% of the scores fall between the mean and minus one.
  • 84.13% are greater than minus one.
  • 15.87% are less than minus one
iq 115 pr
IQ =115, PR = ?
  • z = (115 – 100)/15 = 1.
  • We are above the mean so the answer must be greater than 50%.
  • The answer is 84.13% .
85 iq 115
85 < IQ < 115
  • What percentage of IQ scores fall between 85 (z = -1) and 115 (z = 1)?
  • 34.13% are between mean and -1.
  • 34.13% are between mean and 1.
  • 68.26% are between -1 and 1.
115 iq 130
115 < IQ < 130
  • What percentage of IQ scores fall between 115 (z = 1) and 130 (z = 2)?
  • 84.13% fall below 1.
  • 97.72% fall below 2.
  • 97.72 – 84.13 = 13.59%
the lowest 10
The Lowest 10%
  • What score marks off the lowest 10% of IQ scores ?
  • z = 1.28
  • IQ = 100 – 1.28(15) = 80.8
the middle 50
The Middle 50%
  • What scores mark off the middle 50% of IQ scores?
  • -.67 < z < .67;
  • 100 - .67(15) = 90
  • 100 + .67(15) = 110
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