Introductory Biostatistics

X

ðx
xÞðy
yÞ< 0

for negative association.

With proper standardization, we obtain

r¼

P

ðx
xÞðy
yÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½

P

ðx
xÞ^2 Š½

P

ðy
yÞ^2 Š

q

so that

 1 ara 1

This statistic,r, called thecorrelation coe‰cient, is a popular measure for the
strength of a statistical relationship; here is a shortcut formula:

r¼

P

xy
ð

P

xÞð

P

yÞ=n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½

P

x^2 
ð

P

xÞ^2 =nŠ½

P

y^2 
ð

P

yÞ^2 =nŠ

q

Meanningful interpretation of the correlation coe‰cientris rather compli-
cated at this level. We will revisit the topic in Chapter 8 in the context of
regression analysis, a statistical method that is closely connected to correlation.
Generally:

Values near 1 indicate a strong positive association.

Values near
1 indicate a strong negative association.

Values around 0 indicate a weak association.

Interpretation ofrshould be made cautiously, however. It is true that a
scatter plot of data that results in a correlation number ofþ1or
1 has to lie
in a perfectly straight line. But a correlation of 0 doesn’t mean that there is no
association; it means that there is nolinearassociation. You can have a corre-
lation near 0 and yet have a very strong association, such as the case when the
data fall neatly on a sharply bending curve.

Example 2.8 Consider again the birth-weight problem described earlier in this
section. We have the data given in Table 2.13. Using the five totals, we obtain

r¼

94 ; 322 
½ð 1207 Þð 975 ފ= 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

½ 123 ; 561 
ð 1207 Þ^2 = 12 Š½ 86 ; 487 
ð 975 Þ^2 = 12 Š

q

¼
0 : 946

indicating a very strong negative association.

86 DESCRIPTIVE METHODS FOR CONTINUOUS DATA