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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.



Using the normal curve
Using the Normal Curve size increases.

  • 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 size increases.

  • 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 size increases.

  • Use the normal curve table in our text.

  • Use SPSS or another stats package.

  • Use an Internet resource.


Iq 85 pr
IQ = 85, PR = ? size increases.

  • 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 size increases.

  • 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 | size increases.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 size increases.=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 size increases.

  • 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 size increases.

  • 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% size increases.

  • 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% size increases.

  • What scores mark off the middle 50% of IQ scores?

  • -.67 < z < .67;

  • 100 - .67(15) = 90

  • 100 + .67(15) = 110


Memorize these benchmarks
Memorize These Benchmarks size increases.


The normal distribution1
The Normal Distribution size increases.



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