Final answer:
The Pearson correlation coefficient (r) measures the linear relationship between two variables (x and y), ranging from -1 to +1. It indicates the direction and strength of the relationship. The coefficient of determination (r²), representing the square of r, expresses what proportion of variation in y can be explained by x.
Step-by-step explanation:
The Pearson correlation coefficient, often denoted as r, is a statistical measure developed in the early 1900s by Karl Pearson. It quantifies the linear relationship between two variables, usually labeled as x (independent variable) and y (dependent variable). The value of r ranges from -1 to +1, where 1 means a perfect positive linear relationship, -1 means a perfect negative linear relationship, and 0 denotes no linear relationship at all.
Positive values of r indicate that as x increases, y tends to increase, and vice versa. Negative values suggest that as x increases, y decreases, or the reverse. The closer the value of r is to 1 or -1, the stronger the linear correlation between the variables.
The coefficient is calculated using the formula:
nΣxy -[ΣxΣy]
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√(nΣx² - (Σx)²)(nΣy² - (Σy)²)
Many statistical softwares and calculators can compute this coefficient swiftly. Additionally, the square of r, known as the coefficient of determination, r², is often expressed as a percentage and indicates how much of the variation in the dependent variable y can be explained by the independent variable x.