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.
1
(Y   )2 / 2 2
F (Y ) 
(e)
 2
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
50
68
90
95
98
99
100
Fall Between Plus
and Minus z =
.67
1.
1.645
1.96
2.33
2.58
3.
The Normal Distribution
The Bivariate Normal Distribution