Introductory Biostatistics

3.3 PROBABILITY MODELS FOR CONTINUOUS DATA

In Section 3.2 we treated the family of normal curves very informally because it
was intended to reach more students and readers for whom mathematical for-
mulas may not be very relevant. In this section we provide some supplementary
information that may be desirable for those who may be more interested in the
fundamentals of biostatistical inference.
A class of measurements or a characteristic on which individual observa-
tions or measurements are made is called avariable. If values of a variable may
theoretically lie anywhere on a numerical scale, we have acontinuous variable;
examples include weight, height, and blood pressure, among others. We saw in
Section 3.2 that each continuous variable is characterized by a smoothdensity
curve. Mathematically, a curve can be characterized by an equation of the form

y¼fðxÞ

called aprobability density function, which includes one or several parameters;
the total area under a density curve is 1.0. The probability that the variable
assumes any value in an interval between two specific pointsaandbis given by

ðb

a

fðxÞdx

The probability density function for the family of normal curves, sometimes
referred to as theGaussian distribution, is given by

fðxÞ¼

1

s

ffiffiffiffiffiffi

2 p

p exp

1

2

x
m

s

"# 2

for
y<x<y

The meaning and significance of the parametersmands=s^2 have been dis-
cussed in Section 3.2;mis the mean,s^2 is the variance, andsis the standard
deviation. Whenm¼1 ands^2 ¼1, we have thestandard normal distribution.
The numerical values listed in Appendix B are those given by

ðz

0

1

ffiffiffiffiffiffi

2 p

p exp

1

2

ðxÞ^2



dx

The normal distribution plays an important role in statistical inference
because:

1. Many real-life distributions are approximately normal.


2. Many other distributions can be almost normalized by appropriate data
   transformations (e.g., taking the log). When logX has a normal distri-
   bution,Xis said to have alognormal distribution.

PROBABILITY MODELS FOR CONTINUOUS DATA 131