91Ó°ÊÓ

When we talk about linear transformation, it means changing the form of a variable through specific mathematical operations. Imagine you have a set of numbers, and you multiply each by a constant and then add another constant. That's precisely what a linear transformation does!
In mathematical terms, you can write this as:

• For a given variable \( x_i \), the transformation is \( x'_i = ax_i + c \).
• Similarly, for another variable \( y_i \), it is \( y'_i = by_i + d \).

Here, \( a \) and \( b \) are positive scaling factors that stretch or compress the values, while \( c \) and \( d \) are constants that shift them up or down. A crucial aspect of linear transformations is their impact on relationships between variables, such as correlation.

A fascinating property of the correlation coefficient is its independence from scale. This means that the correlation between two variables doesn't change if you multiply them by positive constants or add constants to them.
Think of it like this: whether you measure temperature in °C or °F, the correlation between temperature and another variable, like ice cream sales, stays the same!
This property arises because, in the calculation of the correlation coefficient, the effects of scaling get canceled out. Whether you magnify all measurements tenfold or shift them a couple of units up, the relative relationship—the correlation—remains unaffected. This is why transformations like those in the exercise, where \( a > 0 \) and \( b > 0 \), do not affect correlation.

The correlation formula gives us a mathematical way to quantify the relationship between two variables. It's like a recipe: follow it properly, and you'll get a result that tells you how closely two variables move together.
The formula is expressed as:

• \( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} \)

Here's how it works:

• \( (x_i - \bar{x}) \) and \( (y_i - \bar{y}) \) are differences between each value and their mean. They show how far each data point is from the average.
• The numerator expresses how these differences multiply together across both variables, showing their joint variation.
• The denominator normalizes this variation by taking into account the separate variations of each variable.

This formula gives \( r \) a value between -1 and 1, where 1 means perfect positive correlation, -1 means perfect negative correlation, and 0 means no correlation.

The proof of a concept in mathematics turns an idea into an undeniable fact. For the independence of the correlation coefficient from the units of measurement, the proof lies in simplification through accurate algebraic manipulations.
As shown in the exercise, you start by transforming the variables \( x_i \) and \( y_i \) into \( x'_i \) and \( y'_i \). Next, substitute these transformed variables into the correlation formula.
The algebraic journey here leads to:

• Using the transformed equations means substituting \( a(x_i - \bar{x}) \) and \( b(y_i - \bar{y}) \) into the formula.
• The scales \( a \) and \( b \) appear in every part of the equation, making both the numerator and denominator larger by the same factor \( ab \).
• By canceling out \( ab \) from both the numerator and denominator, you get back to the original correlation coefficient formula.

This cancellation is crucial, as it shows that regardless of how much you stretch or shrink the scales, the \( r \) value remains unchanged. Thus, this proof confirms that the correlation is indeed scale-independent.