Physics and Engineering of Radiation Detection

9.8. Correlation 559

9.8.A PearsonrorSimpleLinearCorrelation

In essence, the simple linear correlation determines the extent of proportionality
between two variables. The proportionality is quantified through the coefficient
of correlation, which is related to the regression fit of the data. The correlation
coefficient has several equivalent forms but is most conveniently determined from
the relation

r=

N

∑

xiyi−

∑

xi

∑

yi

√[

N

∑

x^2 i−(

∑

xi)^2

][

N

∑

yi^2 −(

∑

yi)^2

], (9.8.1)

where the summation (

∑

) is over the whole dataset belonging to variablesxandy.
The way correlation coefficient is interpreted has already been discussed. Fig.9.8.1
shows a few examples of regression fits to different datasets and the corresponding
correlation coefficients.

y

y y

x

y

x

r~0.9

(a) (b)

r~0.5

r~0.1

(c) (d)

x x

r=−1 r=+1

Figure 9.8.1: A few examples of regression fits to different datasets

and their corresponding correlation coefficients. It should be noted that

r=−1 in (a) represents anti-correlation, which in fact signifies perfect

correlation just liker= 1. As the data get dispersed the correlation

coefficient approaches no-correlation value of 0.