The Normal Distribution

History• Abraham de Moivre (1733) – consultant to

gamblers. Pronunciation.• Pierre Simon Laplace – mathematician,

astronomer, philosopher, determinist.• Carl Friedrich Gauss – mathematician and

astronomer.• Adolphe Quetelet -- mathematician,

astronomer, “social physics.” Pronunciation.

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 BenchmarksThis 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