Master Karl Pearson's Coefficient of Correlation for ungrouped data with CBSE Class 11 Applied Mathematics theory, solved examples, and practice sets.
Karl Pearson's coefficient of correlation, denoted by r, is a quantitative measure of the linear relationship between two variables, X and Y. It ranges from -1 to +1, where +1 indicates a perfect positive linear correlation, -1 indicates a perfect negative linear correlation, and 0 indicates no linear correlation. It is widely used in statistics to determine the strength and direction of the association between two quantitative variables.
Step 1: Calculate means x̄ and ȳ.
Step 2: Find deviations (xᵢ - x̄) and (yᵢ - ȳ).
Step 3: Compute the product of deviations and their squares.
Step 4: Apply the formula.
Answer: For X={1, 2, 3} and Y={2, 4, 6}, r = 1.
For ungrouped data, the computational formula is often preferred as it avoids calculating deviations from the mean, which can involve fractions. This formula utilizes the sums of X, Y, X², Y², and XY. It is mathematically equivalent to the definition formula but more efficient for manual calculation.
Step 1: Given n=3, Σx=6, Σy=12, Σx²=14, Σy²=56, Σxy=28.
Step 2: Substitute into formula: r = [3(28) - (6)(12)] / √[(3(14) - 6²)(3(56) - 12²)].
Step 3: Simplify: r = [84 - 72] / √[(42 - 36)(168 - 144)].
Answer: r = 12 / √[6 · 24] = 12 / 12 = 1.
The correlation coefficient r is independent of the change of origin and scale of the variables. This means if we transform X to U = (X-a)/h and Y to V = (Y-b)/k, the correlation between U and V is the same as between X and Y. It is a dimensionless quantity, meaning it has no units.
Step 1: Given X={10, 20, 30}, Y={100, 200, 300}.
Step 2: Let U = (X-20)/10 = {-1, 0, 1} and V = (Y-200)/100 = {-1, 0, 1}.
Step 3: Calculate r for U and V.
Answer: r = 1.