558 Chapter 9. Essential Statistics for Data Analysis
We are required to obtain equations that can be solved to determine the three
coefficientsα 1 ,α 2 ,andα 3 , This can be done by first constructing the expres-
sion for the sum of the squared residuals, which according to equation 9.7.11
can be written asχ^2 =∑[
α 1 +α 2 xi+α 3 x^2 i−yi] 2
Now, according to equation 9.7.12, the first equation is0=
∂
∂α 1[∑{
α 1 +α 2 xi+α 3 x^2 i−yi} 2 ]
⇒0=2
∑{
α 1 +α 2 xi+α 3 x^2 i−yi}
Similarly the second equation is0=
∂
∂α 2[∑{
α 1 +α 2 xi+α 3 x^2 i−yi} 2 ]
⇒0=2
∑{
α 1 +α 2 xi+α 3 x^2 i−yi}
xi.Andthethirdequationis0=
∂
∂α 3[∑{
α 1 +α 2 xi+α 3 x^2 i−yi} 2 ]
⇒0=2
∑{
α 1 +α 2 xi+α 3 x^2 i−yi}
x^2 i.9.8 Correlation
There are different techniques in statistics that can be used to determine how one
dataset is associated with another one. The specific term used to determine such
association is thecorrelation analysis. An example where such an analysis would
be useful is to see how the change in the leakage current of a silicon detector is
correlated with increase in the absorbed radiation.
The measure of the correlation, no matter what technique is used, always lies
between -1 and +1. A correlation coefficient of +1 signifies perfect correlation
while a value of -1 shows that the data are negatively correlated. Note that negative
correlation does not mean no correlation, rather strong correlation but in an opposite
sense. It would mean that if one variable is increasing the other is decreasing but in a
perfectly correlated manner. A correlation coefficient of 0 represents no correlation.
Even though there are several techniques used to determine correlation, how-
ever the most commonly used technique is the so calledPearsonrorsimple linear
correlation. We will therefore restrict ourselves to this correlation technique.