6.3.2. PROPERTIES OF CORRELATION COEFFICIENT

• r has no unit. It is a pure number. It means units of measurement are not part of r. r between height in feet and weight in kilograms, for instance, could be say 0.7.

The correlation coefficient, represented as 'r', is a statistical measure that describes the strength and direction of a relationship between two variables. Importantly, 'r' does not have any associated units of measurement. This standardization makes it easier to compare correlation coefficients across different pairs of variables, regardless of how those variables are measured. For example, whether you're measuring height in feet or weight in kilograms, a correlation coefficient of 0.7 indicates a similar strength of association between the two variables.

Imagine you are comparing the relationship between the number of hours students study and their grades. When you express both variables in different units (like hours for study and percentages for grades), the correlation remains valid and does not depend on the measurement units, making the results universally understandable.

• A negative value of r indicates an inverse relation. A change in one variable is associated with change in the other variable in the opposite direction. When price of a commodity rises, its demand falls. When the rate of interest rises the demand for funds also falls.

A negative correlation coefficient (r < 0) implies that as one variable increases, the other decreases. This inverse relationship can be seen in various economic situations. For instance, if the price of a product increases, typically the quantity demanded decreases, resulting in a negative correlation between price and demand. This principle is fundamental in economics and helps anticipate how changes in price may affect consumer behavior.

Think about the relationship between the prices of seasonal fruits. If the price of a fruit rises significantly, people might choose to buy less of it or opt for alternatives. Therefore, we see an inverse correlation – as the price climbs, the demand usually diminishes.

• If r is positive the two variables move in the same direction. For instance, when the price of coffee rises, the demand for tea also rises.

A positive correlation coefficient (r > 0) indicates a direct relationship where both variables change in the same direction. For example, if people start to prefer a particular drink (like tea) when the price of coffee increases, the correlation between the price of coffee and tea demand would be positive.

Consider the relationship between exercise and physical fitness. As individuals exercise more (first variable), their fitness level typically increases (second variable). Their correlation would be positive, demonstrating that both variables are enhancing concurrently.

• If r = 1 or r = –1 the correlation is perfect and there is exact linear relation.

Perfect correlation occurs when the correlation coefficient equals 1 or -1, indicating a flawless linear relationship. An r value of 1 shows that as one variable increases, the other increases proportionally, while an r value of -1 indicates that one variable decreases proportionally as the other increases. These cases represent two extremes of predicting one variable from another.

An example of perfect correlation could be found in simple arithmetic: if you have one full cup of coffee and for every cup you add, you drain exactly one cup from another pot. The correlation between coffee consumed and coffee remaining would be perfect, showing the direct dependency between both variables.

• A high value of r indicates a strong linear relationship. Its value is said to be high when it is close to +1 or –1. A low value of r (close to zero) indicates a weak linear relation. But there may be a non-linear relation.

The magnitude of the correlation coefficient conveys the strength of the relationship between variables. A value close to +1 or -1 suggests a strong linear connection, while a value near 0 suggests minimal or no linear relationship. However, it is possible for relationships to exist that are non-linear – meaning the relationship may not follow a straight line but still be significant.

For instance, consider a graph that plots the hours studied versus test scores. A value of 0.9 may suggest that study time and score have a strong positive correlation. Conversely, a value of 0.1 may indicate very little direct relationship; however, this does not rule out other kinds of relationships, such as increased study time leading to improvements at a diminishing rate.

• The value of the correlation coefficient lies between minus one and plus one, –1 ≤ r ≤1.

The correlation coefficient is confined within a range from -1 to +1. Values outside this range would signify errors in calculation or misunderstandings of the data's relationships. A handy rule of thumb for memorizing this range is to remember that -1 indicates a perfect negative correlation while +1 indicates a perfect positive correlation.

Consider throwing darts at a dartboard. A perfect hit in the center distinguishes excellent accuracy (1) while hitting the outermost ring represents a poor throw (-1). If your aim is variable or off-target, this indicates that your throw lacks a direct and consistent correlation to the position of the bullseye.

• The magnitude of r is unaffected by the change of origin and change of scale. Given two variables X and Y let us define two new variables.

Changes in the units or scales of the variables (for example, changing from feet to inches for height) do not impact the correlation coefficient. This means that no matter how we represent or scale our data, the fundamental relationship measured will remain constant.

Think about measuring temperature in Celsius versus Fahrenheit. If you analyze the correlation between temperature and ice cream sales, whether you report temperatures in Celsius or Fahrenheit, the correlation remains stable and informative. The relationship stays intact regardless of how you present the data numerically.