To understand which factor makes the correlation coefficient [tex]\( r \)[/tex] undefined for a vertical line, we need to break down the formula for [tex]\( r \)[/tex] and analyze it step-by-step.
The formula for [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n \sum x y - \sum x \sum y}{\sqrt{\left(n \sum x^2 - \left(\sum x\right)^2\right) \left(n \sum y^2 - \left(\sum y\right)^2\right)}} \][/tex]
We are specifically investigating the case when the data forms a vertical line. This means that all [tex]\( x \)[/tex]-values are the same. Let’s denote this constant value as [tex]\( x_0 \)[/tex].
For a vertical line:
1. Since all [tex]\( x \)[/tex]-values are the same, the sum of [tex]\( x \)[/tex]-values, [tex]\(\sum x\)[/tex], will be [tex]\( n \times x_0 \)[/tex].
2. The sum of [tex]\( x^2 \)[/tex] values, [tex]\(\sum x^2\)[/tex], will be [tex]\( n \times (x_0^2) \)[/tex].
Now let's substitute these back into the term related to [tex]\( x \)[/tex] in the denominator:
[tex]\[ n \sum x^2 - (\sum x)^2 \][/tex]
This becomes:
[tex]\[ n \sum x^2 - (\sum x)^2 = n \times (x_0^2) - (n \times x_0)^2 \][/tex]
Simplify it:
[tex]\[ = n \times (x_0^2) - n^2 \times (x_0^2) \][/tex]
[tex]\[ = (n \times x_0^2) - (n^2 \times x_0^2) \][/tex]
[tex]\[ = n \times x_0^2 \times (1 - n) \][/tex]
Given that [tex]\( x_0 \)[/tex] is a constant and non-zero:
[tex]\[ n \times x_0^2 - n^2 \times x_0^2 = x_0^2 \times (n - n^2) \][/tex]
Simplify further:
[tex]\[ = x_0^2 \times n \times (1 - n) \][/tex]
This simplifies to:
[tex]\[ = x_0^2 \times n \times (1 - n) \][/tex]
From this equation, it's clear that [tex]\( n \times x_0^2 = n \times (x_0)^2 \)[/tex] and [tex]\( n^2 \times (x_0^2) = (n \times x_0)^2 \)[/tex].
Notice that:
[tex]\[ (n \sum x^2 - (\sum x)^2) = x_0^2 \times (n - n^2) = -n n x_0^2 \][/tex]
Finally, the result of this term is zero if:
[tex]\[ 0 = n x_0^2 - n^2 x_0^2 \][/tex]
From this, we see that this denominator term [tex]\( n \sum x^2 - (\sum x)^2 \)[/tex] evaluates to a negative value. In actuality, it makes the denominator zero.
Thus, the problematic term that leads [tex]\( r \)[/tex] to be undefined for a vertical line is:
[tex]\[ \boxed{n \sum x^2 - (\sum x)^2} \][/tex]
