Correlation analysis is the statistical method used to measure the strength and direction of the relationship between two variables. It quantifies how changes in one variable are associated with changes in another. While regression analysis is used to predict the value of a dependent variable based on an independent variable, the fundamental theory for establishing the existence and degree of a relationship between two sets of phenomena is the theory of correlation.

Karl Pearson's coefficient of correlation (r) is the geometric mean of the two regression coefficients, bxy and byx. Mathematically, r = ±√(bxy * byx). This relationship holds because both regression coefficients share the same sign, which determines the sign of the correlation coefficient.

In statistics, the Pearson product-moment correlation coefficient is universally denoted by the lowercase letter 'r'. It measures the strength and direction of a linear relationship between two variables, ranging from -1 to +1.

The coefficient of determination, denoted as R-squared (R²), represents the proportion of the variance in the dependent variable that is predictable from the independent variable. It is calculated as the square of the Pearson correlation coefficient (r). It provides a measure of how well observed outcomes are replicated by the model.

Spearman's rank correlation is a non-parametric measure of rank correlation. It assesses how well the relationship between two variables can be described using a monotonic function. It is specifically designed for ordinal data, where variables are ranked rather than measured on a continuous scale, though it can be applied to interval or ratio data that has been converted to ranks.

The coefficient of determination (r²) represents the proportion of variance in the dependent variable that is predictable from the independent variable. The correlation coefficient (r) is the square root of the coefficient of determination. While r can be negative, r² is always non-negative.

In simple linear regression, the regression line of Y on X is expressed as Y = a + bX. The slope 'b' represents the change in the dependent variable Y for a unit change in the independent variable X. This slope is formally referred to as the regression coefficient of Y on X, denoted as bYX.

A positive correlation implies that as one variable increases, the other variable also tends to increase. In the context of human growth, weight generally increases as age increases during the developmental stages, representing a positive relationship between these two variables.

The correlation coefficient (r) is defined as the geometric mean of the two regression coefficients, denoted as bxy and byx. Mathematically, this is expressed as r = ±√(bxy * byx). Since the geometric mean of two numbers is the square root of their product, this relationship holds true. The sign of the correlation coefficient will match the sign of the regression coefficients.

Since the correlation coefficient r = ±sqrt(bxy * byx), and |r| must be ≤ 1, if one coefficient is > 1, the other must be < 1 to keep the product within the range of -1 to 1. Both statements are mathematically sound.