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.”

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.

• 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

• 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

• 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.

22 2/)()(2

1)(

YeYF

Avoiding the Calculus

• Use the normal curve table in our text.• Use SPSS or another stats package.• Use an Internet resource.

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

• 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

• 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 = ?

• 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

• 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

• 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%

• What score marks off the lowest 10% of IQ scores ?

• z = 1.28• IQ = 100 – 1.28(15) = 80.8

The Middle 50%

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

• -.67 < z < .67; • 100 - .67(15) = 90• 100 + .67(15) = 110

Memorize These Benchmarks

This Middle Percentage of

Scores

Fall Between Plus and Minus z =

50 .6768 1.90 1.64595 1.9698 2.3399 2.58

100 3.

The Normal Distribution

The Bivariate Normal Distribution