The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

54 CHAPTER 4 Normal Distribution and Related Properties

Figure 4.2. Picture of Student ’ s t - distributions (2 and 4 degrees of freedom) and the
standard normal distribution ^.

• df is an abbreviation for degrees of freedom.

Normal

df* = 4

df = 2

unimodal distribution with fatter tails (density drops slower than the

normal), and especially fatter when the degrees of freedom is 5 or less.

See Figure 4.2.

Here is why the t - distribution is important. Our test statistic will

be standard normal when we know the standard deviation and the

observations are normal. But to know what the standard deviation is

equal to is not common in practice. So in place of our test statistic

Zm=−()/(/),μ σ n where m=∑Xni/,

the sample mean, and μ is the population mean, we use

(^) Tm Sn S i Xm ni
n
=− =⎡ − −
⎣⎢

⎤

∑= ⎦⎥

()/(/),μ where 1 ( )/().^21

This pivotal quantity for testing has a Student ’ s t - distribution with n − 1
degrees of freedom, and T approaches Z as n gets large. These state-
ments hold exactly when the X i s are independent and identically dis-
tributed normal random variables. But it also works for large n for other
distributions thanks to the central limit theorem.