Correlation Coefficient

The correlation coefficient describes the strength and direction of a linear relationship between two variables. A value of +1 indicates a perfect positive linear relationship (as X increases, Y increases), while -1 indicates a perfect negative linear relationship (as X increases, Y decreases). Values between 0 and 1 (or 0 and -1) represent varying strengths of positive or negative linear relationships, respectively. A value close to zero means there's little to no linear correlation.

To calculate the correlation coefficient, a table is created with columns for X, Y, XY (product of X and Y), X squared, and Y squared. Example data points (1-6 for X, 2-14 for Y) are used to populate the table, and the respective products and squares are calculated for each row.

After filling the table, the next step is to sum each column: sum of X (21), sum of Y (48), sum of XY (211), sum of X squared (91), and sum of Y squared (490). These sums are crucial for the correlation coefficient formula.

The formula for the correlation coefficient (r) involves the number of data points (n, which is 6 in this example) and the sums calculated previously. The numerator is n multiplied by the sum of XY, minus the product of the sum of X and the sum of Y. The denominator is the square root of [(n times sum of X squared minus (sum of X) squared) multiplied by (n times sum of Y squared minus (sum of Y) squared)].

Plugging in the calculated sums and 'n' into the formula: the numerator becomes (6 * 211) - (21 * 48) = 1266 - 1008 = 258. The denominator involves (6 * 91) - (21 squared) = 546 - 441 = 105, and (6 * 490) - (48 squared) = 2940 - 2304 = 636. Multiplying 105 by 636 gives 66780. Thus, r = 258 / sqrt(66780).

The final calculation yields an r-value of approximately 0.998. This indicates a very strong positive linear relationship between the X and Y variables, meaning as X increases, Y also increases significantly.